Solution 1 to problem over

Equations

The following unsolved equations remain:

     2      2
0=a12  + a13

Expressions

The solution is given through the following expressions:


       - m1*n3*r214
r10=-----------------
     a12*n2 + a13*n3


       - m1*n2*r214
r11=-----------------
     a12*n2 + a13*n3


       - m1*n1*r214
r12=-----------------
     a12*n2 + a13*n3


r13=0


r14=0


r15=0


r20=0


r21=0


r23=0


r24=0


r25=0


r26=0


r27=0


r28=0


r29=r214


r210=0


r212=0


r213=0


       - 2*a13*m1*r214
r215=------------------
      a12*n2 + a13*n3


       - 2*a12*m1*r214
r216=------------------
      a12*n2 + a13*n3


r217=0


r219=0


r220=r214


m3=0


m2=0


c33=0


c23=0


c22=0


c13=0


c12=0


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:

 r214, m1, n1, n2, n3, 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.

 
{a12,r214,a13,a12*n2 + a13*n3}

Relevance for the application:

Modulo the following equation:

     2      2
0=a12  + a13


the system of equations related to the Hamiltonian HAM:

HAM=2*u1*u2*a12 + 2*u1*u3*a13 + u1*n1 + u2*n2 + u3*n3 + v1*m1

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

     2
FI=u1 *(a12*n2 + a13*n3) - 2*u1*v2*a12*m1 - 2*u1*v3*a13*m1

        2                       2
    + u2 *(a12*n2 + a13*n3) + u3 *(a12*n2 + a13*n3) - v1*m1*n1 - v2*m1*n2

    - v3*m1*n3

{HAM,FI} = 0





And again in machine readable form:



HAM=2*u1*u2*a12 + 2*u1*u3*a13 + u1*n1 + u2*n2 + u3*n3 + v1*m1$

FI=u1**2*(a12*n2 + a13*n3) - 2*u1*v2*a12*m1 - 2*u1*v3*a13*m1 + u2**2*(a12*n2 + 
a13*n3) + u3**2*(a12*n2 + a13*n3) - v1*m1*n1 - v2*m1*n2 - v3*m1*n3$