Solution 5 to problem over

Equations

The following unsolved equations remain:

     2      2
0=a22  + a23

Expressions

The solution is given through the following expressions:


r22

   1     4        1     2    2        1     2                 1     3
  ---*a22 *r21 - ---*a22 *a23 *r21 + ---*a22 *a23*b33*r217 + ---*a23 *b33*r217
   2              2                   4                       4
=------------------------------------------------------------------------------
                                       3
                                    a22 *a23


r23=0


r24=0


         1     2            1     2
      - ---*a22 *a23*r21 + ---*a23 *b33*r217
         2                  4
r25=-----------------------------------------
                         3
                      a22


        2           2
     a22 *r217 - a23 *r217
r26=-----------------------
                2
             a22


      - a23*r217
r27=-------------
         a22


r28=0


r29=0


       - a23*r217
r210=-------------
          a22


r212=0


r213=0


r215=0


r216=0


r218=0


r219=0


r220=0


     a23*c23
c33=---------
       a22


                   1     2
      - a23*c23 + ---*b33
                   4
c22=-----------------------
              a22


c13=0


c12=0


      - a23*b33
b32=------------
        a22


b31=0


b13=0


b12=0


b11=b33


        2
     a23
a33=------
     a22

Parameters

Apart from the condition that they must not vanish to give a non-trivial solution and a non-singular solution with non-vanishing denominators, the following parameters are free:

 r21, r217, c23, b33, a22, a23

Inequalities

In the following not identically vanishing expressions are shown. Any auxiliary variables g00?? are used to express that at least one of their coefficients must not vanish, e.g. g0019*p4 + g0020*p3 means that either p4 or p3 or both are non-vanishing.

 
{a22,a23,r217}

Relevance for the application:

Modulo the following equation:

     2      2
0=a22  + a23


the system of equations related to the Hamiltonian HAM:

                           2    2                         2    2
HAM=(4*u1*v1*a22*b33 + 4*u2 *a22  + 8*u2*u3*a22*a23 + 4*u3 *a23

                                              2                    2
      - 4*u3*v2*a23*b33 + 4*u3*v3*a22*b33 + v2 *( - 4*a23*c23 + b33 )

                              2
      + 8*v2*v3*a22*c23 + 4*v3 *a23*c23)/(4*a22)

has apart from the Hamiltonian and Casimirs the following 2 first integrals: 

              3              2                  2
FI=4*u1*v1*a22  - 4*u2*v3*a22 *a23 - 4*u3*v2*a22 *a23

                  3            2      2    2         2     2          2
    + u3*v3*(4*a22  - 4*a22*a23 ) + v1 *a23 *b33 + v2 *(a22 *b33 + a23 *b33)

{HAM,FI} = 0



        2    2     2     2      2
FI= - v1 *a23  + v2 *(a22  - a23 ) + 2*v2*v3*a22*a23

{HAM,FI} = 0





And again in machine readable form:



HAM=(4*u1*v1*a22*b33 + 4*u2**2*a22**2 + 8*u2*u3*a22*a23 + 4*u3**2*a23**2 - 4*u3*
v2*a23*b33 + 4*u3*v3*a22*b33 + v2**2*( - 4*a23*c23 + b33**2) + 8*v2*v3*a22*c23 +
 4*v3**2*a23*c23)/(4*a22)$

FI=4*u1*v1*a22**3 - 4*u2*v3*a22**2*a23 - 4*u3*v2*a22**2*a23 + u3*v3*(4*a22**3 - 
4*a22*a23**2) + v1**2*a23**2*b33 + v2**2*(a22**2*b33 + a23**2*b33)$

FI= - v1**2*a23**2 + v2**2*(a22**2 - a23**2) + 2*v2*v3*a22*a23$