Solution 2 to problem over

Equations

The following unsolved equations remain:

       2      2
0=4*a23  + a33

Expressions

The solution is given through the following expressions:


      1
     ---*a33*r338
      2
r31=--------------
         a23


r32=r37


r33=0


r34=r313


r35=0


r36=0


r38=0


r39=0


       1
      ---*a33*r320
       2
r310=--------------
          a23


       1
      ---*a33*r338
       2
r311=--------------
          a23


r312=0


r314=0


r315=0


                  1
      a23*r330 + ---*a33*r320
                  2
r316=-------------------------
                a23


r317=0


r318=0


r319=0


r321=r338


r322=0


          1
       - ---*a33*r313
          2
r323=-----------------
            a23


r324=0


r325=0


r326=r320


r327=0


r328=0


r329=0


r331=0


r332=0


r333=0


r334=0


r335=0


       1
      ---*a33*r313
       2
r336=--------------
          a23


r337=0


r339=0


r340=0


r341=0


r342=0


r343=0


r344=0


r345=0


r346=0


r347=0


r348=0


r349=0


r350=r330


r351=0


r352=0


r353=0


r354=0


r355=0


c33=0


c23=0


b33=a33


b12=0


b22=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:

 r30, r37, r313, r320, r338, r330, 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.

 
{a23,a33}

Relevance for the application:

Modulo the following equation:

       2      2
0=4*a23  + a33


the system of equations related to the Hamiltonian HAM:

                                  2
HAM=2*u2*u3*a23 + 2*u2*v3*a23 + u3 *a33 + u3*v3*a33

has apart from the Hamiltonian and Casimirs the following 6 first integrals: 

     2        2        2
FI=u1 *v3 + u2 *v3 + u3 *v3

                                        2     2     2
  = a product of the elements of: {v3,u1  + u2  + u3 }

{HAM,FI} = 0



                                  1                  1       2
FI=u1*v1*v3*a23 + u2*v2*v3*a23 + ---*u3*v2*v3*a33 + ---*v2*v3 *a33
                                  2                  2

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

   v3,

   2*u1*v1*a23 + 2*u2*v2*a23 + u3*v2*a33 + v2*v3*a33}

{HAM,FI} = 0



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

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

   v3,

   u3 + v3,

   2*u2*a23 + u3*a33}

{HAM,FI} = 0



    1                  1                                     2
FI=---*u1*v2*v3*a33 - ---*u2*v1*v3*a33 + u3*v1*v3*a23 + v1*v3 *a23
    2                  2

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

   v3,

   u1*v2*a33 - u2*v1*a33 + 2*u3*v1*a23 + 2*v1*v3*a23}

{HAM,FI} = 0



     2        2
FI=v1 *v3 + v2 *v3

                                        2     2
  = a product of the elements of: {v3,v1  + v2 }

{HAM,FI} = 0



     3
FI=v3

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

{HAM,FI} = 0





And again in machine readable form:



HAM=2*u2*u3*a23 + 2*u2*v3*a23 + u3**2*a33 + u3*v3*a33$

FI=u1**2*v3 + u2**2*v3 + u3**2*v3$

FI=u1*v1*v3*a23 + u2*v2*v3*a23 + 1/2*u3*v2*v3*a33 + 1/2*v2*v3**2*a33$

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

FI=1/2*u1*v2*v3*a33 - 1/2*u2*v1*v3*a33 + u3*v1*v3*a23 + v1*v3**2*a23$

FI=v1**2*v3 + v2**2*v3$

FI=v3**3$