%This is "program13_3.red"   :
%computes a local chart for the differentiable manifold  $\matfrak{X}_{M5}$
%in some neighborhood of the canonical CS $J_0$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%  OUTPUT   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 off echo$     off nat$
 OUT  "rprogram13_3.tex"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 ON RAT$
 OFF MSG$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%% Loading the commutation relations file  %%%%%%%%%%%%%%
DIM:= 6$
  in "6nilp/m.5"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%writing the commutation relations
WRITE "\documentclass{article}"$
WRITE "\usepackage{amsmath,amssymb}"$
WRITE "\sloppy"$
WRITE "\begin{document}"$
WRITE "This output from the file \texttt{program13\_3.red}.\\"$
WRITE "Computation of a local chart at the canonical structure on the real Lie Algebra"$
write "USD {\mathcal{G}}_{", dim, ",", PART(REFALGTEX,1), "}.USD"$
WRITE "\smallskip  \par "$
WRITE "Commutation relations for"$
write "USD {\mathcal{G}}_{", dim,",", PART(REFALGTEX,1), "}:USD\\"$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO X_(i,j):=X(i)*x(j)$
%FOR j:=1:DIM DO X(j):=MKID(x_,j)$
FOR i:=1:DIM-1 DO FOR j:=i+1:DIM DO
    IF X_(i,j) NEQ 0 THEN WRITE
 %       "USD[x_",i,",x_",j,"]=", X_(i,j),"USD;"$
        "USD[x(",i,"),x(",j,")]=", X_(i,j),"USD;"$
WRITE "\P"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 operator x,f,A$
 noncom x,f$
 for all j let
  F(X(j))=FOR j1:=1:DIM SUM xi(j1,j)*X(j1)$
  let f(0)=0$
 for all j let f(-x(j))=-f(x(j))$
 FOR ALL J,U SUCH THAT NUMBERP(U) LET f(U*X(j))=U*f(X(j))$
 FOR ALL J,U,V SUCH THAT NUMBERP(U) AND NUMBERP(V) LET f(U*V*X(j))=U*V*f(X(j))$
 FOR ALL J,U,V SUCH THAT NUMBERP(U) AND NUMBERP(V) LET f(-U*V*X(j))=-U*V*f(X(j))$
 FOR ALL s,i,j,k,l
 LET f(X(s) *xi(i,j)*xi(k,l))=xi(i,j)*xi(k,l)*f(X(s))$
 FOR ALL s,i,j,k,l
 LET f(xi(i,j)*xi(k,l)*X(s))=xi(i,j)*xi(k,l)*f(X(s))$
 FOR ALL s,i,j,k,l
 LET f(-X(s) *xi(i,j)*xi(k,l))=-xi(i,j)*xi(k,l)*f(X(s))$
 FOR ALL s,i,j
 LET f(X(s) *xi(i,j))=(xi(i,j))*f(X(s))$
 FOR ALL s,i,j
 LET f(-X(s) *xi(i,j))=-(xi(i,j))*f(X(s))$
 FOR ALL s,i,j
 LET f(-X(s) *xi(i,j)**2)=(-xi(i,j)**2)*f(X(s))$
 FOR ALL s,i,j
 LET f(X(s) *xi(i,j)**2)=(xi(i,j)**2)*f(X(s))$
 FOR ALL s,i,j,k,l
 LET f(-xi(i,j)*xi(k,l)*X(s))=-xi(i,j)*xi(k,l)*f(X(s))$
 FOR ALL J,U SUCH THAT NUMBERP(U) LET f(-U*X(j))=-U*f(X(j))$
 FOR ALL A,B LET F(A+B)=F(A)+F(B)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 for all j,k let
  temp110(j,k) = x(j)*F(x(k))$
 for all j,k let
  temp120(j,k) = F(temp110(j,k))$
 for all j,k let
  temp101(j,k) = F(x(j))*x(k)$
 for all j,k let
  temp102(j,k) = F(temp101(j,k))$
 for all j,k let
  temp2(j,k) =    F(x(j))* F(x(k)) $
 for all j,k let  temp0(j,k) =  x(j)*x(k)$
 for all j,k let
  A(j,k)=  -temp2(j,k) + temp0(j,k) + temp120(j,k) + temp102(j,k);

COLLECT_TORSION:=FOR j1:=1:DIM-1 JOIN
   FOR j2:=j1+1:DIM JOIN
   IF A(j1,j2) NEQ 0 THEN
   {{{j1,j2},A(j1,j2)}}
   ELSE {}$

%IF LENGTH(COLLECT_TORSION) NEQ 0 THEN WRITE "Nonzero torsion
%   in the following cases :",COLLECT_TORSION

IF LENGTH(COLLECT_TORSION) NEQ 0 THEN WRITE "Nonzero torsion"
ELSE WRITE "Zero torsion"$

write "\par"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%off exp$
%on factor$

%la liste des équations non nulles
%write "list of the nonzero torsion equations"$
COLLECT_EQ:=FOR j1:=1:LENGTH(COLLECT_TORSION) JOIN
   FOR j2:=1:DIM JOIN
   IF V(j2)*PART(PART(COLLECT_TORSION,j1),2) NEQ 0 THEN
   {{{PART(PART(COLLECT_TORSION,j1),1),j2}, V(j2)*PART(PART(COLLECT_TORSION,j1),2)} }
    ELSE {}$

%COMMENT
% WRITE "Torsion equations to cancel (Reduce output) : \\", COLLECT_EQ$

% WRITE "Torsion equations to cancel (Latex output) : \\USD"$
% for each A in COLLECT_EQ do
%        if PART(A,2) neq 0 then
%        <<write PART(PART(A,1),1),"|", PART(PART(A,1),2),"\\",PART(A,2),"\\">>   $
%        %<<write "\\", A,"\\">>   $
% write "USD"$


 off factor$
 on exp$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%% writing the specific file containing the reduce data for J %%%%%%%%%%%%%
out "matJ13_3.red"$
WRITE "% This file contains the REDUCE data for a chart in a neighborhood of $J_0$"$
WRITE "% and the torsion equation to be canceled"$
write "% on the Lie algebra g_{", dim, ",", PART(REFALGTEX,1), "}"$

WRITE "matrix J(6,6)"$
%WRITE "J:=",MATJ$
WRITE "% Writing the torsion equation in Reduce format:"$
FOR i:=1:DIM DO FOR  k:=1:DIM DO << xi(i,k):=J(i,k)>>$
%write "COLLECT_EQ:=",COLLECT_EQ$
write "PROCEDURE COLLECT_EQ$",COLLECT_EQ$
for i:=1:DIM do for k:=1:DIM do <<CLEAR xi(i,k)>>$
 out "rprogram13_3.tex"$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par Simultaneous resolution of the nonzero torsion equations and the matrix"$
write "equation USD J^2 = -I . USD"$
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Suppose first USDxi(1,5) \neq 0USD."$
write "\\ Then one first gets"$
write "\\ from equation USD45|1USD :"$
write "USDxi(1,6):=-xi(2,5)USD"$
write "\\ and then from equation USD36|2USD : USD xi(2,6)**2 = -xi(2,5)**2USD, hence"$
write "USDxi(2,6):=0,xi(2,5):=0,xi(1,6):=0USD"$
write "But then from equation USD35|1USD : USD xi(1,5)=0USD  a contradiction"$
write "Hence USD xi(1,5)USD has to be USD 0 USD"$
xi(1,5):=0$
write "\\ USD xi(1,5):=", xi(1,5),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "then, one gets from  equation USD45|2USD :"$
xi(2,5):=0$
write "\\ USD xi(2,5):=", xi(2,5),"USD"$
write "\\ and from equation USD46|1USD :"$
xi(1,6):=0$
write "\\ USD xi(1,6):=", xi(1,6),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "then, one gets from  equation USD36|2USD :"$
xi(2,6):=0$
write "\\ USD xi(2,6):=", xi(2,6),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Suppose now USDxi(4,6) \neq 0USD."$
write "\\ Then one first gets"$
write "\\ from equation USD26|4USD : which reads USD (xi(4,5)-xi(3,6))xi(4,6)=0USD"$
write "USDxi(4,5):=xi(3,6)USD"$
write "\\ and then from equation USD16|4USD"$
write "which reads USD xi(4,6)$**2 + xi(4,5)xi(3,6)=0USD"$
write ": USD xi(4,6)**2 = -xi(3,6)**2USD, hence"$
write "USDxi(4,6):=0,xi(3,6):=0USD"$
write "  a contradiction"$
write "Hence USD xi(4,6)USD has to be USD 0 USD"$
xi(4,6):=0$
write "\\ USD xi(4,6):=", xi(4,6),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "then, one gets from  equation USD25|4USD :"$
xi(4,5):=0$
write "\\ USD xi(4,5):=", xi(4,5),"USD"$
write "\\ and from equation USD26|6USD :"$
xi(3,6):=0$
write "\\ USD xi(3,6):=", xi(3,6),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "then, one gets from  equation USD15|3USD :"$
xi(3,5):=0$
write "\\ USD xi(3,5):=", xi(3,5),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 5x5 entry in  USD J**2 USD  is "$
write "USD  {J**2}^5_5=xi(5,5)**2 + xi(6,5)*xi(5,6);USD\\"$
write "Hence USD xi(5,6)xi(6,5)\neq 0 USD"$
write "and USD xi(5,6) =(-1-xi(5,5)**2)/xi(6,5). USD"$
xi(5,6):= (-1-xi(5,5)**2)/xi(6,5)$
write "\\ USD xi(5,6):=", ws,"USD"$
write "Moreover, the 5x6 entry in  USD J**2 USD  is "$
write "USD  (xi(5,5) + xi(6,6))*xi(5,6);USD hence\\"$
xi(6,6):= -xi(5,5)$
write "\\ USD xi(6,6):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 6x1 entry in  USD J**2 USD  gives "$

%USD  J**2^6_1=xi(6,5)*xi(5,1) + xi(6,4)*xi(4,1) + xi(6,3)*xi(3,1) + xi(6,2)*xi(2
%,1) - xi(6,1)*xi(5,5) + xi(6,1)*xi(1,1);USD\\$
xi(5,1):=-(
+ xi(6,4)*xi(4,1) + xi(6,3)*xi(3,1) + xi(6,2)*xi(2
,1) - xi(6,1)*xi(5,5) + xi(6,1)*xi(1,1))/xi(6,5)$
write "\\ USD xi(5,1):=", ws,"USD"$

write "and the 6x2 entry in  USD J**2 USD  gives "$
%USD  J**2^6_2=xi(6,5)*xi(5,2) + xi(6,4)*xi(4,2) + xi(6,3)*xi(3,2) - xi(6,2)*xi(5
%,5) + xi(6,2)*xi(2,2) + xi(6,1)*xi(1,2);USD\\$
xi(5,2):=-(
+ xi(6,4)*xi(4,2) + xi(6,3)*xi(3,2) - xi(6,2)*xi(5
,5) + xi(6,2)*xi(2,2) + xi(6,1)*xi(1,2))/xi(6,5)$
write "\\ USD xi(5,2):=", ws,"USD"$

write "and the 6x3 entry in  USD J**2 USD  gives "$
%USD  J**2^6_3=xi(6,5)*xi(5,3) + xi(6,4)*xi(4,3) - xi(6,3)*xi(5,5) + xi(6,3)*xi(3
%,3) + xi(6,2)*xi(2,3) + xi(6,1)*xi(1,3);USD\\$
xi(5,3):=-(
+ xi(6,4)*xi(4,3) - xi(6,3)*xi(5,5) + xi(6,3)*xi(3
,3) + xi(6,2)*xi(2,3) + xi(6,1)*xi(1,3))/xi(6,5)$
write "\\ USD xi(5,3):=", ws,"USD"$

write "and the 6x4 entry in  USD J**2 USD  gives "$
%USD  J**2^6_4=xi(6,5)*xi(5,4) - xi(6,4)*xi(5,5) + xi(6,4)*xi(4,4) + xi(6,3)*xi(3
%,4) + xi(6,2)*xi(2,4) + xi(6,1)*xi(1,4);USD\\$
xi(5,4) :=-(- xi(6,4)*xi(5,5) + xi(6,4)*xi(4,4) + xi(6,3)*xi(3
,4) + xi(6,2)*xi(2,4) + xi(6,1)*xi(1,4))/xi(6,5)$
write "\\ USD xi(5,4):=", ws,"USD"$
write "From the trace of USD JUSD, we get:"$
xi(4,4):=-for i:=1:3 sum xi(i,i)$
write "\\ USD xi(4,4):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ \textbf{Case} USDxi(2,1) \neq 0 \\USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 1x1 entry in  USD J**2 USD  gives "$
%USD  J**2^1_1=xi(4,1)*xi(1,4) + xi(3,1)*xi(1,3) + xi(2,1)*xi(1,2) + xi(1,1)**2
%;USD\\$
xi(1,2):=(-1-(xi(4,1)*xi(1,4) + xi(3,1)*xi(1,3) + xi(1,1)**2))/xi(2,1)$
write "\\ USD xi(1,2):=", ws,"USD"$


write "and the 2x1 entry in  USD J**2 USD  gives "$
%USD  J**2^2_1=xi(4,1)*xi(2,4) + xi(3,1)*xi(2,3) + xi(2,2)*xi(2,1) + xi(2,1)*xi(1
%,1);USD\\$
xi(1,1):=-(xi(4,1)*xi(2,4) + xi(3,1)*xi(2,3) + xi(2,2)*xi(2,1))/xi(2,1)$
write "\\ USD xi(1,1):=", ws,"USD"$

write "and the 2x3 entry in  USD J**2 USD  gives "$
%USD  J**2^2_3=xi(4,3)*xi(2,4) + xi(3,3)*xi(2,3) + xi(2,3)*xi(2,2) + xi(2,1)*xi(1,3);USD\\$
xi(1,3):=-(xi(4,3)*xi(2,4) + xi(3,3)*xi(2,3) + xi(2,3)*xi(2,2))/xi(2,1)$
write "\\ USD xi(1,3):=", ws,"USD"$

write "and the 2x4 entry in  USD J**2 USD  gives "$
%USD  J**2^2_4=xi(3,4)*xi(2,3) - xi(3,3)*xi(2,4) - xi(2,4)*xi(1,1) + xi(2,1)*xi(1,4);USD\\$
xi(1,4):=-(xi(3,4)*xi(2,3) - xi(3,3)*xi(2,4) - xi(2,4)*xi(1,1))/xi(2,1)$
write "\\ USD xi(1,4):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 4x1 entry in  USD J**2 USD  gives "$

%USD  J**2^4_1=xi(4,3)*xi(3,1) + xi(4,2)*xi(2,1) - xi(4,1)*xi(3,3) - xi(4,1)*xi(2
%,2);USD\\$
xi(4,2):=-(xi(4,3)*xi(3,1)  - xi(4,1)*xi(3,3) - xi(4,1)*xi(2,2))/xi(2,1)$

write "\\ USD xi(4,2):=", ws,"USD"$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\par With these values, \textit{Reduce} computes again all equations."$
write "Then the 3x1 entry in  USD J**2 USD  gives "$

%SD  J**2^3_1=(xi(4,1)*xi(3,4)*xi(2,1) - xi(4,1)*xi(3,1)*xi(2,4) + xi(3,3)*xi(3,
%)*xi(2,1) + xi(3,2)*xi(2,1)**2 - xi(3,1)**2*xi(2,3) - xi(3,1)*xi(2,2)*xi(2,1))/
%i(2,1);USD\\$
xi(3,2):=-(xi(4,1)*xi(3,4)*xi(2,1) - xi(4,1)*xi(3,1)*xi(2,4) + xi(3,3)*xi(3,
1)*xi(2,1)
%+ xi(3,2)*xi(2,1)**2
- xi(3,1)**2*xi(2,3) - xi(3,1)*xi(2,2)*xi(2,1))/xi(2,1)**2$
write "\\ USD xi(3,2):=", ws,"USD\\"$

write "The 4x4 entry in  USD J**2 USD  reads "$
%USD  J**2^4_4=(xi(4,3)*xi(3,4)*xi(2,1)**2 - xi(4,3)*xi(3,1)*xi(2,4)*xi(2,1) - xi
%(4,1)*xi(3,4)*xi(2,3)*xi(2,1) + xi(4,1)*xi(3,1)*xi(2,4)*xi(2,3) + xi(3,3)**2*xi(
%2,1)**2 - 2*xi(3,3)*xi(3,1)*xi(2,3)*xi(2,1) + xi(3,1)**2*xi(2,3)**2)/xi(2,1)**2
%;USD\\$
write "USD(xi(4,3)*xi(2,1)- xi(4,1)*xi(2,3))(xi(3,4)*xi(2,1) -xi(3,1)*xi(2,4))"$
write "+(xi(3,3)*xi(2,1) - xi(3,1)*xi(2,3))**2)/xi(2,1)**2 =-1USD"$
write "Hence the case USD(xi(3,4)*xi(2,1)- xi(3,1)*xi(2,4)) =0USD is impossible"$
write "Then we get :"$
xi(4,3):= (((-xi(2,1)**2 - (xi(3,3)*xi(2,1)- xi(3,1)*xi(2,3))**2)
/(xi(3,4)*xi(2,1) -xi(3,1)*xi(2,4)) ) +xi(4,1)*xi(2,3)) /xi(2,1)$
write "\\ USD xi(4,3):=", ws,"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "At this stage, we definitely have USDJ^2= -1 USD."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 WRITE "Torsion equations to cancel (Latex output) : \\USD"$
 for each A in COLLECT_EQ do
        if PART(A,2) neq 0 then
        <<write PART(PART(A,1),1),"|", PART(PART(A,1),2),"\\",PART(A,2),"\\">>   $
        %<<write "\\", A,"\\">>   $
write "USD"$
write "The remaining equations to be solved amount to solving the 2 following:"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 for each A in COLLECT_EQ do
       <<if PART(A,1) ={{1,4},5} then
         f145:=PART(A,2)  else
        if PART(A,1) ={{1,4},6} then
         f146:=PART(A,2)>> $

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "Equation 14|5 : f145:=0 , with f145:=", f145$
write "Equation 14|6 : f146:=0 , with f146:=", f146$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% The following solves the system
write "\\ From equation USD14|5USD : "$
%{{{1,4},5},
%(xi(6,5)*xi(5,5)*xi(3,4)*xi(2,1)**2 + xi(6,5)*xi(5,5)*xi(2,1)**3 + xi(6,5)*xi(4,
%1)*xi(3,4)*xi(2,4)*xi(2,1) - xi(6,5)*xi(4,1)*xi(3,1)*xi(2,4)**2 + xi(6,5)*xi(3,4
%)*xi(2,2)*xi(2,1)**2 + xi(6,5)*xi(3,3)*xi(3,1)*xi(2,4)*xi(2,1) + xi(6,5)*xi(3,3)
%*xi(2,1)**3 - xi(6,5)*xi(3,1)**2*xi(2,4)*xi(2,3) - xi(6,5)*xi(3,1)*xi(2,4)*xi(2,
%2)*xi(2,1) - xi(6,5)*xi(3,1)*xi(2,3)*xi(2,1)**2 + xi(5,5)**2*xi(3,3)*xi(2,1)**2
%+ xi(5,5)**2*xi(2,2)*xi(2,1)**2 + xi(3,3)*xi(2,1)**2 + xi(2,2)*xi(2,1)**2)/(xi(6
%,5)*xi(2,1)**2)},


xi(3,3):=-
(xi(6,5)*xi(5,5)*xi(3,4)*xi(2,1)**2 + xi(6,5)*xi(5,5)*xi(2,1)**3 + xi(6,5)*xi(4,
1)*xi(3,4)*xi(2,4)*xi(2,1) - xi(6,5)*xi(4,1)*xi(3,1)*xi(2,4)**2 + xi(6,5)*xi(3,4
)*xi(2,2)*xi(2,1)**2
%+ xi(6,5)*xi(3,3)*xi(3,1)*xi(2,4)*xi(2,1) + xi(6,5)*xi(3,3)*xi(2,1)**3
- xi(6,5)*xi(3,1)**2*xi(2,4)*xi(2,3) - xi(6,5)*xi(3,1)*xi(2,4)*xi(2,
2)*xi(2,1) - xi(6,5)*xi(3,1)*xi(2,3)*xi(2,1)**2
%+ xi(5,5)**2*xi(3,3)*xi(2,1)**2
+ xi(5,5)**2*xi(2,2)*xi(2,1)**2
%+ xi(3,3)*xi(2,1)**2
+ xi(2,2)*xi(2,1)**2)
/(xi(6,5)*xi(2,1)*(xi(3,1)*xi(2,4)+xi(2,1)**2)+xi(2,1)**2*(1+xi(5,5)**2))$

write "\\ USD xi(3,3):=", ws,"USD"$
write "\\ USD DEN xi(3,3):=", DEN xi(3,3),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\\ Then from equation USD14|6USD : "$
%{{{1,4},6},

xi(3,4):=-
(
%xi(6,5)**2*xi(3,4)*xi(3,1)*xi(2,4)*xi(2,1) + xi(6,5)**2*xi(3,4)*xi(2,1)**3

+ xi(6,5)**2*xi(3,1)*xi(2,4)*xi(2,1)**2 + xi(6,5)**2*xi(2,1)**4

%- xi(6,5)*xi(5,5)*xi(4,1)*xi(3,4)*xi(2,4)*xi(2,1)

+ xi(6,5)*xi(5,5)*xi(4,1)*xi(3,1)*xi(2,4)**2

%+ xi(6,5)*xi(5,5)*xi(3,4)*xi(3,1)*xi(2,3)*xi(2,1)

+ xi(6,5)*xi(5,5)*xi(3,1)**2*xi(2,4
)*xi(2,3) + 2*xi(6,5)*xi(5,5)*xi(3,1)*xi(2,4)*xi(2,2)*xi(2,1) + 2*xi(6,5)*xi(5,5
)*xi(3,1)*xi(2,3)*xi(2,1)**2 + 2*xi(6,5)*xi(5,5)*xi(2,2)*xi(2,1)**3

%+ xi(6,5)*xi(4,1)*xi(3,4)*xi(2,4)*xi(2,2)*xi(2,1)- xi(6,5)*xi(4,1)*xi(3,4)*xi(2,3)*xi(2,1)**2

- xi(6,5)*xi(4,1)*xi(3,1)*xi(2,4)**2*xi(2,2) + xi(6,5)*xi(4,1)*xi(3,1)*xi(2,
4)*xi(2,3)*xi(2,1)

%+ xi(6,5)*xi(3,4)*xi(3,1)*xi(2,4)*xi(2,1)**2 + xi(6,5)*xi(3,4)*xi(3,1)*xi(2,3)*xi(2,2)*xi(2,1)


%+ xi(6,5)*xi(3,4)*xi(2,2)**2*xi(2,1)**2 + xi(6,5)*xi(3,4)*xi(2,1)**4 + xi(6,5)*xi(3,4)*xi(2,1)**2

- xi(6,5)*xi(3,1)**2*xi(2,4)**2*xi(2,1)

- xi(6,5)*xi(3,1)**2*xi(2,4)*xi(2,3)*xi(2,2) - xi(6,5)*xi(3,1)*xi(2,
4)*xi(2,2)**2*xi(2,1) - xi(6,5)*xi(3,1)*xi(2,4)*xi(2,1)**3 + xi(6,5)*xi(3,1)*xi(
2,4)*xi(2,1) + 2*xi(6,5)*xi(2,1)**3
%- xi(5,5)**2*xi(4,1)*xi(3,4)*xi(2,3)*xi(2,1)

+ xi(5,5)**2*xi(4,1)*xi(3,1)*xi(2,4)*xi(2,3)
%+ xi(5,5)**2*xi(3,4)*xi(2,1)**3

+ xi(5,5)**2*xi(3,1)**2*xi(2,3)**2 - xi(5,5)**2*xi(3,1)*xi(2,4)*xi(2,1)**2 + 2*xi(
5,5)**2*xi(3,1)*xi(2,3)*xi(2,2)*xi(2,1) + xi(5,5)**2*xi(2,2)**2*xi(2,1)**2 + xi(
5,5)**2*xi(2,1)**2
%- xi(4,1)*xi(3,4)*xi(2,3)*xi(2,1)

+ xi(4,1)*xi(3,1)*xi(2,4)*xi(2,3)
%+ xi(3,4)*xi(2,1)**3

+ xi(3,1)**2*xi(2,3)**2 - xi(3,1)*xi(2,4)*xi(2,1)**
2 + 2*xi(3,1)*xi(2,3)*xi(2,2)*xi(2,1) + xi(2,2)**2*xi(2,1)**2 + xi(2,1)**2)/
(
xi(6,5)**2*xi(3,1)*xi(2,4)*xi(2,1) + xi(6,5)**2*xi(2,1)**3
- xi(6,5)*xi(5,5)*xi(4,1)*xi(2,4)*xi(2,1)
+ xi(6,5)*xi(5,5)*xi(3,1)*xi(2,3)*xi(2,1)
+ xi(6,5)*xi(4,1)*xi(2,4)*xi(2,2)*xi(2,1)- xi(6,5)*xi(4,1)*xi(2,3)*xi(2,1)**2
+ xi(6,5)*xi(3,1)*xi(2,4)*xi(2,1)**2 + xi(6,5)*xi(3,1)*xi(2,3)*xi(2,2)*xi(2,1)
+ xi(6,5)*xi(2,2)**2*xi(2,1)**2 + xi(6,5)*xi(2,1)**4 + xi(6,5)*xi(2,1)**2
- xi(5,5)**2*xi(4,1)*xi(2,3)*xi(2,1)
+ xi(5,5)**2*xi(2,1)**3
- xi(4,1)*xi(2,3)*xi(2,1)
+ xi(2,1)**3)$
write "\\ USD xi(3,4):=", ws,"USD"$
write "\\ USD DEN xi(3,4):=", DEN xi(3,4),"USD"$
write "in the neighborhood of any point such that the two preceding denominators"$
write "do not vanish."$
write "this is not the case of the canonical J_0."$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%off exp$
%on factor$

WRITE "USD \par Now the nonzero torsion equations left are :"$
write COLLECT_EQ$


 MATRIX MATJ(DIM,DIM)$
 FOR i:=1:DIM DO FOR j:=1:DIM DO MATJ(i,j):=xi(i,j)$

off exp$
on factor$
 FOR i:=1:DIM DO FOR j:=1:DIM DO
 <<WRITE "USD  J^",i,"_",j,"=",MATJ(i,j),";USD\\">>$

on exp$
off factor$
 %%%%%%%%%%%%%%%%%
 MATJCARRE:=(MATJ)**2$

 %WRITE "Matrice de J**2:=",(MATJ)**2$


  FOR i:=1:DIM DO FOR j:=1:DIM DO
  <<WRITE "USD  {J**2}^",i,"_",j,"=",MATJCARRE(i,j),";USD\\">>$

 WRITE "\\$ det J:=",DET(MATJ)$
 WRITE "Trace J:=",TRACE(MATJ)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "We now solve the system in the neigborhood of J_0."$
write "clear xi(3,3),xi(3,4)"$
clear xi(3,3),xi(3,4)$

%update matJ

 MATRIX MATJ(DIM,DIM)$
 FOR i:=1:DIM DO FOR j:=1:DIM DO MATJ(i,j):=xi(i,j)$

off exp$
on factor$
 FOR i:=1:DIM DO FOR j:=1:DIM DO
 <<WRITE "USD  J^",i,"_",j,"=",MATJ(i,j),";USD\\">>$

on exp$
off factor$
 %%%%%%%%%%%%%%%%%
 MATJCARRE:=(MATJ)**2$

 %WRITE "Matrice de J**2:=",(MATJ)**2$


  FOR i:=1:DIM DO FOR j:=1:DIM DO
  <<WRITE "USD  {J**2}^",i,"_",j,"=",MATJCARRE(i,j),";USD\\">>$

 WRITE "\\$ det J:=",DET(MATJ)$
 WRITE "Trace J:=",TRACE(MATJ)$


WRITE "USD \par Now the nonzero torsion equations left are :"$
write COLLECT_EQ$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
factor xi(5,5),xi(6,5)$
write "f145=",f145$
write "f146=",f146$
write "\\ From equation USD14|6USD : "$

xi(6,5):=-(
xi(5,5)*(xi(3,3) + xi(2,2)) - xi(4,1)*xi(3,4)
*xi(2,3)*xi(2,1)**(-1) + xi(4,1)*xi(3,1)*xi(2,4)*xi(2,3)*xi(2,1)**(-2) + xi(3,4)
*xi(2,1) - xi(3,3)*xi(3,1)*xi(2,3)*xi(2,1)**(-1) - xi(3,3)*xi(2,2) + xi(3,1)**2*
xi(2,3)**2*xi(2,1)**(-2) - xi(3,1)*xi(2,4) + xi(3,1)*xi(2,3)*xi(2,2)*xi(2,1)**(
-1) + 1 )/
(xi(3,4) + xi(2,1)) $
write "\\ USD xi(6,5):=", xi(6,5),"USD"$
write "\\ USD DEN xi(6,5):=", DEN xi(6,5),"USD"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "the equation USD 14|5USD reads"$
factor xi(5,5)$
write "f145=",f145$
write "NUMf145=",NUM(f145)$
write "DENf145=",DEN(f145)$
write "and gives:"$
xi(5,5):=

-(- xi(4,1)**2*xi(3,4)**2*xi(2,4)*xi(2,3)*xi(2,1)**2 + 2*xi(4,1)**2*xi(3,4)*xi(
3,1)*xi(2,4)**2*xi(2,3)*xi(2,1) - xi(4,1)**2*xi(3,1)**2*xi(2,4)**3*xi(2,3) + xi(
4,1)*xi(3,4)**2*xi(2,4)*xi(2,1)**4 - xi(4,1)*xi(3,4)**2*xi(2,3)*xi(2,2)*xi(2,1)
**3 - 2*xi(4,1)*xi(3,4)*xi(3,3)*xi(3,1)*xi(2,4)*xi(2,3)*xi(2,1)**2 - xi(4,1)*xi(
3,4)*xi(3,3)*xi(2,4)*xi(2,2)*xi(2,1)**3 - xi(4,1)*xi(3,4)*xi(3,3)*xi(2,3)*xi(2,1
)**4 + 2*xi(4,1)*xi(3,4)*xi(3,1)**2*xi(2,4)*xi(2,3)**2*xi(2,1) - 2*xi(4,1)*xi(3,
4)*xi(3,1)*xi(2,4)**2*xi(2,1)**3 + 3*xi(4,1)*xi(3,4)*xi(3,1)*xi(2,4)*xi(2,3)*xi(
2,2)*xi(2,1)**2 + xi(4,1)*xi(3,4)*xi(3,1)*xi(2,3)**2*xi(2,1)**3 + xi(4,1)*xi(3,4
)*xi(2,4)*xi(2,1)**3 + 2*xi(4,1)*xi(3,3)*xi(3,1)**2*xi(2,4)**2*xi(2,3)*xi(2,1) +
 xi(4,1)*xi(3,3)*xi(3,1)*xi(2,4)**2*xi(2,2)*xi(2,1)**2 + xi(4,1)*xi(3,3)*xi(3,1)
*xi(2,4)*xi(2,3)*xi(2,1)**3 - 2*xi(4,1)*xi(3,1)**3*xi(2,4)**2*xi(2,3)**2 + xi(4,
1)*xi(3,1)**2*xi(2,4)**3*xi(2,1)**2 - 2*xi(4,1)*xi(3,1)**2*xi(2,4)**2*xi(2,3)*xi
(2,2)*xi(2,1) - xi(4,1)*xi(3,1)**2*xi(2,4)*xi(2,3)**2*xi(2,1)**2 - xi(4,1)*xi(3,
1)*xi(2,4)**2*xi(2,1)**2 + xi(3,4)**2*xi(2,2)*xi(2,1)**5 + xi(3,4)*xi(3,3)*xi(3,
1)*xi(2,4)*xi(2,1)**4 - xi(3,4)*xi(3,3)*xi(3,1)*xi(2,3)*xi(2,2)*xi(2,1)**3 - xi(
3,4)*xi(3,3)*xi(2,2)**2*xi(2,1)**4 + xi(3,4)*xi(3,3)*xi(2,1)**6 - xi(3,4)*xi(3,3
)*xi(2,1)**4 - xi(3,4)*xi(3,1)**2*xi(2,4)*xi(2,3)*xi(2,1)**3 + xi(3,4)*xi(3,1)**
2*xi(2,3)**2*xi(2,2)*xi(2,1)**2 - 2*xi(3,4)*xi(3,1)*xi(2,4)*xi(2,2)*xi(2,1)**4 +
 xi(3,4)*xi(3,1)*xi(2,3)*xi(2,2)**2*xi(2,1)**3 - xi(3,4)*xi(3,1)*xi(2,3)*xi(2,1)
**5 - xi(3,3)**2*xi(3,1)**2*xi(2,4)*xi(2,3)*xi(2,1)**2 - xi(3,3)**2*xi(3,1)*xi(2
,4)*xi(2,2)*xi(2,1)**3 - xi(3,3)**2*xi(3,1)*xi(2,3)*xi(2,1)**4 - xi(3,3)**2*xi(2
,2)*xi(2,1)**5 + 2*xi(3,3)*xi(3,1)**3*xi(2,4)*xi(2,3)**2*xi(2,1) - xi(3,3)*xi(3,
1)**2*xi(2,4)**2*xi(2,1)**3 + 3*xi(3,3)*xi(3,1)**2*xi(2,4)*xi(2,3)*xi(2,2)*xi(2,
1)**2 + 2*xi(3,3)*xi(3,1)**2*xi(2,3)**2*xi(2,1)**3 + xi(3,3)*xi(3,1)*xi(2,4)*xi(
2,2)**2*xi(2,1)**3 - xi(3,3)*xi(3,1)*xi(2,4)*xi(2,1)**5 + xi(3,3)*xi(3,1)*xi(2,4
)*xi(2,1)**3 + 2*xi(3,3)*xi(3,1)*xi(2,3)*xi(2,2)*xi(2,1)**4 - xi(3,1)**4*xi(2,4)
*xi(2,3)**3 + xi(3,1)**3*xi(2,4)**2*xi(2,3)*xi(2,1)**2 - 2*xi(3,1)**3*xi(2,4)*xi
(2,3)**2*xi(2,2)*xi(2,1) - xi(3,1)**3*xi(2,3)**3*xi(2,1)**2 + xi(3,1)**2*xi(2,4)
**2*xi(2,2)*xi(2,1)**3 - xi(3,1)**2*xi(2,4)*xi(2,3)*xi(2,2)**2*xi(2,1)**2 + xi(3
,1)**2*xi(2,4)*xi(2,3)*xi(2,1)**4 - xi(3,1)**2*xi(2,4)*xi(2,3)*xi(2,1)**2 - xi(3
,1)**2*xi(2,3)**2*xi(2,2)*xi(2,1)**3 - xi(3,1)*xi(2,4)*xi(2,2)*xi(2,1)**3 - xi(3
,1)*xi(2,3)*xi(2,1)**4 - xi(2,2)*xi(2,1)**5
)/
(xi(2,1)**2*( - xi(4,1)*xi(3,4)**2*xi(2,3)*xi(2,1) + xi(4,1)*xi(3
,4)*xi(3,3)*xi(2,4)*xi(2,1) + xi(4,1)*xi(3,4)*xi(3,1)*xi(2,4)*xi(2,3) + xi(4,1)*
xi(3,4)*xi(2,4)*xi(2,2)*xi(2,1) - xi(4,1)*xi(3,4)*xi(2,3)*xi(2,1)**2 - xi(4,1)*
xi(3,3)*xi(3,1)*xi(2,4)**2 - xi(4,1)*xi(3,1)*xi(2,4)**2*xi(2,2) + xi(4,1)*xi(3,1
)*xi(2,4)*xi(2,3)*xi(2,1) + xi(3,4)**2*xi(2,1)**3 - xi(3,4)*xi(3,3)*xi(3,1)*xi(2
,3)*xi(2,1) + xi(3,4)*xi(3,1)**2*xi(2,3)**2 - xi(3,4)*xi(3,1)*xi(2,4)*xi(2,1)**2
 + xi(3,4)*xi(3,1)*xi(2,3)*xi(2,2)*xi(2,1) + xi(3,4)*xi(2,2)**2*xi(2,1)**2 + xi(
3,4)*xi(2,1)**4 + xi(3,4)*xi(2,1)**2 + xi(3,3)**2*xi(3,1)*xi(2,4)*xi(2,1) + xi(3
,3)**2*xi(2,1)**3 - xi(3,3)*xi(3,1)**2*xi(2,4)*xi(2,3) - 2*xi(3,3)*xi(3,1)*xi(2,
3)*xi(2,1)**2 - xi(3,1)**2*xi(2,4)*xi(2,3)*xi(2,2) + xi(3,1)**2*xi(2,3)**2*xi(2,
1) - xi(3,1)*xi(2,4)*xi(2,2)**2*xi(2,1) - xi(3,1)*xi(2,4)*xi(2,1)**3 + xi(2,1)**
3))$
write "\\ USD xi(5,5):=", ws,"USD"$
write "\\ USD DEN xi(5,5):=", DEN xi(5,5),"USD"$
write "f145=",f145$
write "f146=",f146$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%off exp$
%on factor$

%WRITE "USD \par Now the nonzero torsion equations left are :"$

%write COLLECT_EQ$
%%%%%%%%%%%%%%%% writing the file for J %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
out "matJ13_3.red"$
WRITE "% Writing the entries of J:"$
WRITE "% J(i,k) denotes the entry in the ith row , kth column"$
WRITE "% i.e. stands for  the  LaTex expression J^i_k"$
WRITE "% Similarly, xi(i,k) stands for the  LaTex expression \xi^i_k"$

write "%clear xi(5,5)"$
clear xi(5,5)$
write "%clear xi(6,5)"$
clear xi(6,5)$
 FOR i:=1:DIM DO FOR j:=1:DIM DO
 <<WRITE "J(",i,",",j,"):=",MATJ(i,j)>>$
 write "%where "$

xi(5,5):=
-(- xi(4,1)**2*xi(3,4)**2*xi(2,4)*xi(2,3)*xi(2,1)**2 + 2*xi(4,1)**2*xi(3,4)*xi(
3,1)*xi(2,4)**2*xi(2,3)*xi(2,1) - xi(4,1)**2*xi(3,1)**2*xi(2,4)**3*xi(2,3) + xi(
4,1)*xi(3,4)**2*xi(2,4)*xi(2,1)**4 - xi(4,1)*xi(3,4)**2*xi(2,3)*xi(2,2)*xi(2,1)
**3 - 2*xi(4,1)*xi(3,4)*xi(3,3)*xi(3,1)*xi(2,4)*xi(2,3)*xi(2,1)**2 - xi(4,1)*xi(
3,4)*xi(3,3)*xi(2,4)*xi(2,2)*xi(2,1)**3 - xi(4,1)*xi(3,4)*xi(3,3)*xi(2,3)*xi(2,1
)**4 + 2*xi(4,1)*xi(3,4)*xi(3,1)**2*xi(2,4)*xi(2,3)**2*xi(2,1) - 2*xi(4,1)*xi(3,
4)*xi(3,1)*xi(2,4)**2*xi(2,1)**3 + 3*xi(4,1)*xi(3,4)*xi(3,1)*xi(2,4)*xi(2,3)*xi(
2,2)*xi(2,1)**2 + xi(4,1)*xi(3,4)*xi(3,1)*xi(2,3)**2*xi(2,1)**3 + xi(4,1)*xi(3,4
)*xi(2,4)*xi(2,1)**3 + 2*xi(4,1)*xi(3,3)*xi(3,1)**2*xi(2,4)**2*xi(2,3)*xi(2,1) +
 xi(4,1)*xi(3,3)*xi(3,1)*xi(2,4)**2*xi(2,2)*xi(2,1)**2 + xi(4,1)*xi(3,3)*xi(3,1)
*xi(2,4)*xi(2,3)*xi(2,1)**3 - 2*xi(4,1)*xi(3,1)**3*xi(2,4)**2*xi(2,3)**2 + xi(4,
1)*xi(3,1)**2*xi(2,4)**3*xi(2,1)**2 - 2*xi(4,1)*xi(3,1)**2*xi(2,4)**2*xi(2,3)*xi
(2,2)*xi(2,1) - xi(4,1)*xi(3,1)**2*xi(2,4)*xi(2,3)**2*xi(2,1)**2 - xi(4,1)*xi(3,
1)*xi(2,4)**2*xi(2,1)**2 + xi(3,4)**2*xi(2,2)*xi(2,1)**5 + xi(3,4)*xi(3,3)*xi(3,
1)*xi(2,4)*xi(2,1)**4 - xi(3,4)*xi(3,3)*xi(3,1)*xi(2,3)*xi(2,2)*xi(2,1)**3 - xi(
3,4)*xi(3,3)*xi(2,2)**2*xi(2,1)**4 + xi(3,4)*xi(3,3)*xi(2,1)**6 - xi(3,4)*xi(3,3
)*xi(2,1)**4 - xi(3,4)*xi(3,1)**2*xi(2,4)*xi(2,3)*xi(2,1)**3 + xi(3,4)*xi(3,1)**
2*xi(2,3)**2*xi(2,2)*xi(2,1)**2 - 2*xi(3,4)*xi(3,1)*xi(2,4)*xi(2,2)*xi(2,1)**4 +
 xi(3,4)*xi(3,1)*xi(2,3)*xi(2,2)**2*xi(2,1)**3 - xi(3,4)*xi(3,1)*xi(2,3)*xi(2,1)
**5 - xi(3,3)**2*xi(3,1)**2*xi(2,4)*xi(2,3)*xi(2,1)**2 - xi(3,3)**2*xi(3,1)*xi(2
,4)*xi(2,2)*xi(2,1)**3 - xi(3,3)**2*xi(3,1)*xi(2,3)*xi(2,1)**4 - xi(3,3)**2*xi(2
,2)*xi(2,1)**5 + 2*xi(3,3)*xi(3,1)**3*xi(2,4)*xi(2,3)**2*xi(2,1) - xi(3,3)*xi(3,
1)**2*xi(2,4)**2*xi(2,1)**3 + 3*xi(3,3)*xi(3,1)**2*xi(2,4)*xi(2,3)*xi(2,2)*xi(2,
1)**2 + 2*xi(3,3)*xi(3,1)**2*xi(2,3)**2*xi(2,1)**3 + xi(3,3)*xi(3,1)*xi(2,4)*xi(
2,2)**2*xi(2,1)**3 - xi(3,3)*xi(3,1)*xi(2,4)*xi(2,1)**5 + xi(3,3)*xi(3,1)*xi(2,4
)*xi(2,1)**3 + 2*xi(3,3)*xi(3,1)*xi(2,3)*xi(2,2)*xi(2,1)**4 - xi(3,1)**4*xi(2,4)
*xi(2,3)**3 + xi(3,1)**3*xi(2,4)**2*xi(2,3)*xi(2,1)**2 - 2*xi(3,1)**3*xi(2,4)*xi
(2,3)**2*xi(2,2)*xi(2,1) - xi(3,1)**3*xi(2,3)**3*xi(2,1)**2 + xi(3,1)**2*xi(2,4)
**2*xi(2,2)*xi(2,1)**3 - xi(3,1)**2*xi(2,4)*xi(2,3)*xi(2,2)**2*xi(2,1)**2 + xi(3
,1)**2*xi(2,4)*xi(2,3)*xi(2,1)**4 - xi(3,1)**2*xi(2,4)*xi(2,3)*xi(2,1)**2 - xi(3
,1)**2*xi(2,3)**2*xi(2,2)*xi(2,1)**3 - xi(3,1)*xi(2,4)*xi(2,2)*xi(2,1)**3 - xi(3
,1)*xi(2,3)*xi(2,1)**4 - xi(2,2)*xi(2,1)**5
)/
(xi(2,1)**2*( - xi(4,1)*xi(3,4)**2*xi(2,3)*xi(2,1) + xi(4,1)*xi(3
,4)*xi(3,3)*xi(2,4)*xi(2,1) + xi(4,1)*xi(3,4)*xi(3,1)*xi(2,4)*xi(2,3) + xi(4,1)*
xi(3,4)*xi(2,4)*xi(2,2)*xi(2,1) - xi(4,1)*xi(3,4)*xi(2,3)*xi(2,1)**2 - xi(4,1)*
xi(3,3)*xi(3,1)*xi(2,4)**2 - xi(4,1)*xi(3,1)*xi(2,4)**2*xi(2,2) + xi(4,1)*xi(3,1
)*xi(2,4)*xi(2,3)*xi(2,1) + xi(3,4)**2*xi(2,1)**3 - xi(3,4)*xi(3,3)*xi(3,1)*xi(2
,3)*xi(2,1) + xi(3,4)*xi(3,1)**2*xi(2,3)**2 - xi(3,4)*xi(3,1)*xi(2,4)*xi(2,1)**2
 + xi(3,4)*xi(3,1)*xi(2,3)*xi(2,2)*xi(2,1) + xi(3,4)*xi(2,2)**2*xi(2,1)**2 + xi(
3,4)*xi(2,1)**4 + xi(3,4)*xi(2,1)**2 + xi(3,3)**2*xi(3,1)*xi(2,4)*xi(2,1) + xi(3
,3)**2*xi(2,1)**3 - xi(3,3)*xi(3,1)**2*xi(2,4)*xi(2,3) - 2*xi(3,3)*xi(3,1)*xi(2,
3)*xi(2,1)**2 - xi(3,1)**2*xi(2,4)*xi(2,3)*xi(2,2) + xi(3,1)**2*xi(2,3)**2*xi(2,
1) - xi(3,1)*xi(2,4)*xi(2,2)**2*xi(2,1) - xi(3,1)*xi(2,4)*xi(2,1)**3 + xi(2,1)**
3))$
write "xi(5,5):=",xi(5,5)$



write "%and"$

xi(6,5):=-(
xi(5,5)*(xi(3,3) + xi(2,2)) - xi(4,1)*xi(3,4)
*xi(2,3)*xi(2,1)**(-1) + xi(4,1)*xi(3,1)*xi(2,4)*xi(2,3)*xi(2,1)**(-2) + xi(3,4)
*xi(2,1) - xi(3,3)*xi(3,1)*xi(2,3)*xi(2,1)**(-1) - xi(3,3)*xi(2,2) + xi(3,1)**2*
xi(2,3)**2*xi(2,1)**(-2) - xi(3,1)*xi(2,4) + xi(3,1)*xi(2,3)*xi(2,2)*xi(2,1)**(
-1) + 1 )/
(xi(3,4) + xi(2,1)) $


write "xi(6,5):=", xi(6,5)$

write "%to check the torsion equation"$
write "COLLECT_EQ();%"$

write "%to check J**2"$
write "write J**2:=J**2"$


write "%for the suitable  values of the parameters, J=J_0"$
write "xi(2,1):=",-1$
write "xi(2,2):=",0$
write "xi(2,3):=",0$
write "xi(2,4):=",0$
write "xi(3,1):=",0$
write "xi(3,3):=",0$
write "xi(3,4):=",-1$
write "xi(4,1):=",0$
write "xi(6,1):=",0$
write "xi(6,2):=",0$
write "xi(6,3):=",0$
write "xi(6,4):=",0$
%write "write J:=",J$
write "write J"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "clear xi(2,1),xi(2,2),xi(2,3),xi(2,4),xi(3,1),xi(3,3),xi(3,4),xi(4,1)"$
write "clear xi(6,1),xi(6,2),xi(6,3),xi(6,4)"$
%write "write J:=",matJ$
 write "END"$
 shut "matJ13_3.red"$
 out "rprogram13_3.tex"$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 MATRIX MATJ(DIM,DIM)$
 FOR i:=1:DIM DO FOR j:=1:DIM DO MATJ(i,j):=xi(i,j)$

off exp$
on factor$
 FOR i:=1:DIM DO FOR j:=1:DIM DO
 <<WRITE "USD  J^",i,"_",j,"=",MATJ(i,j),";USD\\">>$

on exp$
off factor$
 %%%%%%%%%%%%%%%%%
 MATJCARRE:=(MATJ)**2$

 %WRITE "Matrice de J**2:=",(MATJ)**2$


  FOR i:=1:DIM DO FOR j:=1:DIM DO
  <<WRITE "USD  {J**2}^",i,"_",j,"=",MATJCARRE(i,j),";USD\\">>$

 WRITE "\\$ det J:=",DET(MATJ)$
 WRITE "Trace J:=",TRACE(MATJ)$


WRITE "USD \par Now the nonzero torsion equations left are :"$
write COLLECT_EQ$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
write "\end{document}"$
bye$