Solution 1 to problem over

Equations

The following unsolved equations remain:

     2      2
0=a22  + a23

Expressions

The solution is given through the following expressions:


      - 2*a23*c13*r14
r10=------------------
         a22*b13


      - 2*c13*r14
r11=--------------
         b13


         1
      - ---*a23*b13*r14
         2
r12=--------------------
               2
            a22


     a23*r14
r13=---------
       a22


r15=0


    a22*m3
m2=--------
     a23


        1
     - ---*b13*n3
        2
m1=---------------
         a22


    a22*n3
n2=--------
     a23


n1=0


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


        4        1     2        2    1     3    2
     a22 *c23 - ---*a22 *a23*b13  - ---*a23 *b13
                 4                   4
c22=----------------------------------------------
                          3
                       a22 *a23


      - a23*c13
c12=------------
        a22


b33=0


b32=0


           2          2
      - a22 *b13 - a23 *b13
b31=------------------------
                 2
              a22


      - a23*b13
b12=------------
        a22


b11=0


        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:

 r14, m3, c13, c23, n3, b13, 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.

 
{b13,a22,a23,r14,{m3,n3}}

Relevance for the application:

Modulo the following equation:

     2      2
0=a22  + a23


the system of equations related to the Hamiltonian HAM:

                 2    2                3             2    4
HAM=( - u1*v2*a22 *a23 *b13 + u1*v3*a22 *a23*b13 + u2 *a22 *a23

                   3    2         4        2    2    3
      + 2*u2*u3*a22 *a23  + u2*a22 *n3 + u3 *a22 *a23

                     3                  3              3
      + u3*v1*( - a22 *a23*b13 - a22*a23 *b13) + u3*a22 *a23*n3

                   2    2                  3            1        2
      - 2*v1*v2*a22 *a23 *c13 + 2*v1*v3*a22 *a23*c13 - ---*v1*a22 *a23*b13*n3
                                                        2

          2     4        1     2        2    1     3    2
      + v2 *(a22 *c23 - ---*a22 *a23*b13  - ---*a23 *b13 )
                         4                   4

                   3                 4        2    2    2             3
      + 2*v2*v3*a22 *a23*c23 + v2*a22 *m3 + v3 *a22 *a23 *c23 + v3*a22 *a23*m3)/

        3
    (a22 *a23)

has apart from the Hamiltonian and Casimirs only the following first integral: 

         2                         1            2           2
FI=u2*a22 *b13 + u3*a22*a23*b13 - ---*v1*a23*b13  - 2*v2*a22 *c13
                                   2

    - 2*v3*a22*a23*c13

{HAM,FI} = 0





And again in machine readable form:



HAM=( - u1*v2*a22**2*a23**2*b13 + u1*v3*a22**3*a23*b13 + u2**2*a22**4*a23 + 2*u2
*u3*a22**3*a23**2 + u2*a22**4*n3 + u3**2*a22**2*a23**3 + u3*v1*( - a22**3*a23*
b13 - a22*a23**3*b13) + u3*a22**3*a23*n3 - 2*v1*v2*a22**2*a23**2*c13 + 2*v1*v3*
a22**3*a23*c13 - 1/2*v1*a22**2*a23*b13*n3 + v2**2*(a22**4*c23 - 1/4*a22**2*a23*
b13**2 - 1/4*a23**3*b13**2) + 2*v2*v3*a22**3*a23*c23 + v2*a22**4*m3 + v3**2*a22
**2*a23**2*c23 + v3*a22**3*a23*m3)/(a22**3*a23)$

FI=u2*a22**2*b13 + u3*a22*a23*b13 - 1/2*v1*a23*b13**2 - 2*v2*a22**2*c13 - 2*v3*
a22*a23*c13$