Solution 2 to problem over

Equations

The following unsolved equations remain:

     2      2
0=a23  + a33

Expressions

The solution is given through the following expressions:


        2            1     2
     a23 *a33*r11 + ---*a33 *b33*r14
                     2
r10=---------------------------------
                     3
                  a23


r12=0


     a33*r14
r13=---------
       a23


r15=0


       2       1
    a23 *m3 - ---*a33*b33*n3
               2
m2=--------------------------
            a23*a33


m1=0


    a23*n3
n2=--------
     a33


n1=0


     a33*c23
c33=---------
       a23


        3        1     2    2
     a23 *c23 + ---*a33 *b33
                 4
c22=--------------------------
                2
             a23 *a33


c13=0


c12=0


      - a33*b33
b32=------------
        a23


b31=0


b13=0


b12=0


b11=b33


        2
     a23
a22=------
     a33

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:

 r11, r14, m3, c23, n3, b33, a33, 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.

 
                 2       1
{r14,a23,a33,{a23 *m3 - ---*a33*b33*n3,m3,n3}}
                         2

Relevance for the application:

Modulo the following equation:

     2      2
0=a23  + a33


the system of equations related to the Hamiltonian HAM:

              2             2    4              3             3
HAM=(u1*v1*a23 *a33*b33 + u2 *a23  + 2*u2*u3*a23 *a33 + u2*a23 *n3

          2    2    2                2                2                 2
      + u3 *a23 *a33  - u3*v2*a23*a33 *b33 + u3*v3*a23 *a33*b33 + u3*a23 *a33*n3

          2     3        1     2    2               2
      + v2 *(a23 *c23 + ---*a33 *b33 ) + 2*v2*v3*a23 *a33*c23
                         4

               3       1                      2        2             2
      + v2*(a23 *m3 - ---*a23*a33*b33*n3) + v3 *a23*a33 *c23 + v3*a23 *a33*m3)/(
                       2

          2
       a23 *a33)

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

         3         2        1        2
FI=u2*a23  + u3*a23 *a33 + ---*v3*a33 *b33
                            2

{HAM,FI} = 0



FI=v2*a23 + v3*a33

{HAM,FI} = 0





And again in machine readable form:



HAM=(u1*v1*a23**2*a33*b33 + u2**2*a23**4 + 2*u2*u3*a23**3*a33 + u2*a23**3*n3 + 
u3**2*a23**2*a33**2 - u3*v2*a23*a33**2*b33 + u3*v3*a23**2*a33*b33 + u3*a23**2*
a33*n3 + v2**2*(a23**3*c23 + 1/4*a33**2*b33**2) + 2*v2*v3*a23**2*a33*c23 + v2*(
a23**3*m3 - 1/2*a23*a33*b33*n3) + v3**2*a23*a33**2*c23 + v3*a23**2*a33*m3)/(a23
**2*a33)$

FI=u2*a23**3 + u3*a23**2*a33 + 1/2*v3*a33**2*b33$

FI=v2*a23 + v3*a33$