Solution 3 to problem over

Equations

The following unsolved equations remain:

       2      2
0=4*a23  + b33

Expressions

The solution is given through the following expressions:


r11=0


r12=0


r13=0


r14=0


r15=0


         1     2
      - ---*b33 *r27
         4
r21=-----------------
             2
          a23


r22=0


r23=r28


r24=0


r26=0


r29=0


r210=0


          1
       - ---*b33*r27
          2
r211=----------------
           a23


          1
       - ---*b33*r28
          2
r212=----------------
           a23


r214=0


r215=0


       1
      ---*b33*r28
       2
r216=-------------
          a23


          1
       - ---*b33*r27
          2
r217=----------------
           a23


r218=0


r219=0


m3=0


n3=0


c33=0


c23=0


b12=0


b22=0


a33=b33

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:

 r00, r10, r20, r27, r28, b33, 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.

 
{a23,b33}

Relevance for the application:

Modulo the following equation:

       2      2
0=4*a23  + b33


the system of equations related to the Hamiltonian HAM:

                                  2
HAM=2*u2*u3*a23 + 2*u2*v3*a23 + u3 *b33 + u3*v3*b33

has apart from the Hamiltonian and Casimirs the following 5 first integrals: 

    1               1
FI=---*u1*v2*b33 - ---*u2*v1*b33 + u3*v1*a23 + v1*v3*a23
    2               2

                                    1
  = a product of the elements of: {---,
                                    2

   u1*v2*b33 - u2*v1*b33 + 2*u3*v1*a23 + 2*v1*v3*a23}

{HAM,FI} = 0



       1                   1                           2    1           2
FI= - ---*u1*v1*a23*b33 - ---*u2*v2*a23*b33 + u3*v2*a23  - ---*v2*v3*b33
       2                   2                                4

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

                                                     2            2
    - 2*u1*v1*a23*b33 - 2*u2*v2*a23*b33 + 4*u3*v2*a23  - v2*v3*b33 }

{HAM,FI} = 0



     2
FI=v3

  = a product of the elements of: {v3,v3}

{HAM,FI} = 0



FI=v3

which the program can not factorize further.

{HAM,FI} = 0



FI=1

which the program can not factorize further.

{HAM,FI} = 0





And again in machine readable form:



HAM=2*u2*u3*a23 + 2*u2*v3*a23 + u3**2*b33 + u3*v3*b33$

FI=1/2*u1*v2*b33 - 1/2*u2*v1*b33 + u3*v1*a23 + v1*v3*a23$

FI= - 1/2*u1*v1*a23*b33 - 1/2*u2*v2*a23*b33 + u3*v2*a23**2 - 1/4*v2*v3*b33**2$

FI=v3**2$

FI=v3$

FI=1$