Solution 3 to problem over

Equations

The following unsolved equations remain:

     2      2
0=a12  + a13

Expressions

The solution is given through the following expressions:


      1
     ---*n3*r215
      2
r10=-------------
         a13


         1
      - ---*n3*r215
         2
r11=----------------
          a12


      1
     ---*n1*r215
      2
r12=-------------
         a13


r13=0


r14=0


r15=0


r20=0


r21=0


r23=0


r24=0


      1
     ---*b11*r215
      4
r25=--------------
         a13


r26=0


r27=0


r28=0


r29=0


r210=0


r212=0


r213=0


r214=0


      a12*r215
r216=----------
        a13


r217=0


r219=0


r220=0


m3=0


m2=0


     - a13*n3
n2=-----------
       a12


c33=c22


c23=0


c13=0


c12=0


b33=0


b31=0


b21=0


b13=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:

 r215, m1, c22, n1, b11, 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.

 
{a13,r215,a12}

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  + 2*u1*u3*a12*a13 + u1*v1*a12*b11 + u1*a12*n1 - u2*a13*n3

                                  2             2
      + u3*a12*n3 + v1*a12*m1 + v2 *a12*c22 + v3 *a12*c22)/a12

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

            2                    1    2            1               1
FI=u1*v2*a12  + u1*v3*a12*a13 + ---*v1 *a12*b11 + ---*v1*a12*n1 - ---*v2*a13*n3
                                 4                 2               2

       1
    + ---*v3*a12*n3
       2

{HAM,FI} = 0





And again in machine readable form:



HAM=(2*u1*u2*a12**2 + 2*u1*u3*a12*a13 + u1*v1*a12*b11 + u1*a12*n1 - u2*a13*n3 + 
u3*a12*n3 + v1*a12*m1 + v2**2*a12*c22 + v3**2*a12*c22)/a12$

FI=u1*v2*a12**2 + u1*v3*a12*a13 + 1/4*v1**2*a12*b11 + 1/2*v1*a12*n1 - 1/2*v2*a13
*n3 + 1/2*v3*a12*n3$