Solution 1 to problem over

Equations

The following unsolved equations remain:

     2        2
0=b12  + 4*c23

Expressions

The solution is given through the following expressions:


         1
      - ---*m3*n3*r28
         2
r00=------------------
         b12*c23


r10=0


         1     2           1     2             2
      - ---*b12 *m3*r28 + ---*b12 *n3*r28 - c23 *m3*r28
         4                 4
r11=----------------------------------------------------
                                 2
                          b12*c23


r12=0


r13=0


      - m3*r28
r14=-----------
        b12


r15=0


r20=0


r21=0


r22=0


r23=0


r24=0


r25=0


      1
     ---*b12*r28
      2
r26=-------------
         c23


r27=0


r29=0


r210=0


       1
      ---*b12*r28
       2
r211=-------------
          c23


r212=0


r213=0


r214=0


r215= - r28


r216=0


       1
      ---*b12*r28
       2
r217=-------------
          c23


r218=0


r219=0


r220=0


c33=0


b33=0


b22=0


a33=0


a23=c23

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:

 r28, m3, n3, b12, c23

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.

 
{b12,r28,c23}

Relevance for the application:

Modulo the following equation:

     2        2
0=b12  + 4*c23


the system of equations related to the Hamiltonian HAM:

HAM=u1*v2*b12 + 2*u2*u3*c23 - u2*v1*b12 + 2*u2*v3*c23 + 2*u3*v2*c23 + u3*n3

     + 2*v2*v3*c23 + v3*m3

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

    1              2                       2    1              2
FI=---*u1*u3*v1*b12 *c23 - u1*u3*v3*b12*c23  + ---*u1*v1*v3*b12 *c23
    2                                           2

           2        2    1              2                2
    - u1*v3 *b12*c23  + ---*u2*u3*v2*b12 *c23 - u2*u3*c23 *m3
                         2

       1              2                2        2           2
    + ---*u2*v2*v3*b12 *c23 - u2*v3*c23 *m3 + u3 *v1*b12*c23
       2

       1    2       2                       2
    + ---*u3 *v3*b12 *c23 + u3*v1*v3*b12*c23
       2

                 1     2       1     2         2        1       2    2
    + u3*v2*( - ---*b12 *m3 + ---*b12 *n3 - c23 *m3) + ---*u3*v3 *b12 *c23
                 4             4                        2

       1                            1     2       1     2         2
    - ---*u3*c23*m3*n3 + v2*v3*( - ---*b12 *m3 + ---*b12 *n3 - c23 *m3)
       2                            4             4

       1
    - ---*v3*c23*m3*n3
       2

                                    1
  = a product of the elements of: {---,
                                    4

   u3 + v3,

              2                      2              2               2
   2*u1*v1*b12 *c23 - 4*u1*v3*b12*c23  + 2*u2*v2*b12 *c23 - 4*u2*c23 *m3

                     2              2
    + 4*u3*v1*b12*c23  + 2*u3*v3*b12 *c23

                2         2           2
    + v2*( - b12 *m3 + b12 *n3 - 4*c23 *m3) - 2*c23*m3*n3}

{HAM,FI} = 0





And again in machine readable form:



HAM=u1*v2*b12 + 2*u2*u3*c23 - u2*v1*b12 + 2*u2*v3*c23 + 2*u3*v2*c23 + u3*n3 + 2*
v2*v3*c23 + v3*m3$

FI=1/2*u1*u3*v1*b12**2*c23 - u1*u3*v3*b12*c23**2 + 1/2*u1*v1*v3*b12**2*c23 - u1*
v3**2*b12*c23**2 + 1/2*u2*u3*v2*b12**2*c23 - u2*u3*c23**2*m3 + 1/2*u2*v2*v3*b12
**2*c23 - u2*v3*c23**2*m3 + u3**2*v1*b12*c23**2 + 1/2*u3**2*v3*b12**2*c23 + u3*
v1*v3*b12*c23**2 + u3*v2*( - 1/4*b12**2*m3 + 1/4*b12**2*n3 - c23**2*m3) + 1/2*u3
*v3**2*b12**2*c23 - 1/2*u3*c23*m3*n3 + v2*v3*( - 1/4*b12**2*m3 + 1/4*b12**2*n3 -
 c23**2*m3) - 1/2*v3*c23*m3*n3$