Solution 5 to problem over

Equations

The following unsolved equations remain:

     2      2
0=a12  + a13

Expressions

The solution is given through the following expressions:


r40=0


r41=0


r42=0


r43=0


r45=0


r46=0


r47=0


r48=0


      1
     ---*c12*r4106
      2
r49=---------------
          a13


r410=0


       1
      ---*c12*r4106
       2
r411=---------------
           a13


r412=0


r413=0


          1     2
       - ---*a12 *c12*r4106
          4
r414=-----------------------
                 3
              a13


r415=0


r416=0


r417=0


r418=0


r419=0


r420=0


r421=0


r422=0


r423=0


r424=0


r425=0


r426=0


r427=0


r428=0


r429=0


r430=0


r431=0


r432=0


r433=0


r434=0


r435=0


r436=0


r437=0


r439=0


r440=0


r441=0


r442=0


r443=0


r444=0


r445=0


r446=0


r447=0


r448=0


r449=0


r450=0


r451=0


r452=0


r453=0


r454=0


r455=0


r456=0


r458=0


r459=0


r460=0


r461=0


r462=0


r463=0


r464=0


r465=0


r466=0


r467=0


r468=0


r469=0


r470=0


r471=0


r472=0


r473=0


r474=0


r475=0


r476=0


r477=0


r478=0


r479=0


r480=0


r481=0


r483=0


r484=0


r485=0


r486=0


r487=0


r488=0


r489=0


r490=0


r491=0


r492=0


r493=0


         2            2
      a12 *r4106 + a13 *r4106
r494=-------------------------
              a12*a13


r495=0


r496=0


r498=0


r499=0


r4100=0


r4101=0


r4102=0


r4103=0


r4104=0


           1
        - ---*a12*r4106
           2
r4105=------------------
             a13


           1
        - ---*a13*r4106
           2
r4107=------------------
             a12


r4108=0


r4109=0


        1     2          1     2
       ---*a12 *r4106 + ---*a13 *r4106
        2                2
r4110=---------------------------------
                   a12*a13


r4111=0


r4112=0


r4113=0


r4115=0


r4116=0


r4117=0


r4118=0


r4119=0


r4120=0


r4121=0


r4122=0


r4123=0


r4124=0


r4125=0


c33=c22


c23=0


     a12*c12
c13=---------
       a13


b33=0


b31=0


b21=0


b13=0


b11=0

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:

 r4106, c22, c12, a12, a13

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.

 
{r4106,a13,a12,c12}

Relevance for the application:

Modulo the following equation:

     2      2
0=a12  + a13


the system of equations related to the Hamiltonian HAM:

                                  2
HAM=(2*u1*u2*a12*a13 + 2*u1*u3*a13  + 2*v1*v2*a13*c12 + 2*v1*v3*a12*c12

          2             2
      + v2 *a13*c22 + v3 *a13*c22)/a13

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

     2   2   1     2    2    1     4     1    2   2    4     2              3
FI=u1 *v1 *(---*a12 *a13  + ---*a13 ) - ---*u1 *v2 *a13  + u1 *v2*v3*a12*a13
             2               2           2

       1    2   2    2    2                   2    2      4     1    4    3
    - ---*u1 *v3 *a12 *a13  + u1*u2*v1*v2*(a12 *a13  + a13 ) - ---*v1 *a12 *c12
       2                                                        4

       1    2   2        2        1    2   2        2
    + ---*v1 *v2 *a12*a13 *c12 + ---*v1 *v3 *a12*a13 *c12
       2                          2

{HAM,FI} = too large to simplify





And again in machine readable form:



HAM=(2*u1*u2*a12*a13 + 2*u1*u3*a13**2 + 2*v1*v2*a13*c12 + 2*v1*v3*a12*c12 + v2**
2*a13*c22 + v3**2*a13*c22)/a13$

FI=u1**2*v1**2*(1/2*a12**2*a13**2 + 1/2*a13**4) - 1/2*u1**2*v2**2*a13**4 + u1**2
*v2*v3*a12*a13**3 - 1/2*u1**2*v3**2*a12**2*a13**2 + u1*u2*v1*v2*(a12**2*a13**2 +
 a13**4) - 1/4*v1**4*a12**3*c12 + 1/2*v1**2*v2**2*a12*a13**2*c12 + 1/2*v1**2*v3
**2*a12*a13**2*c12$