Solution 1 to problem over

Equations

The following unsolved equations remain:

     2      2
0=a22  + b12 *kap

Expressions

The solution is given through the following expressions:


r10=0


r11=0


r12=0


r14=0


r15=0


r20=0


r21=0


r22=0


r23=0


r24=0


r27=0


r28=0


r210=0


      m3*r328
r211=---------
        b12


r212=0


r213=0


r214=0


r215=0


      m3*r328
r217=---------
        b12


r218=0


r219=0


r220=0


r30=0


r31=0


r32=0


r33=0


r34=0


r35=0


r36=0


r37=0


r38=0


r39=0


r310=r315


r311=0


r312=r315


r313=0


r314=0


      b12*r343 + m3*r487
r316=--------------------
             b12


r317=0


r318=0


r320=0


r321=0


r322=0


r323=0


r324=0


r325=0


r326=0


r327=r343


r329=0


r330=0


r331=0


r332=0


       - a22*r328 + b12*kap*r315
r333=----------------------------
                 b12


r334=0


r335=0


r336=0


r337=0


r338=0


r339=0


r340=0


r341=0


r342= - r328


r344=0


r345=0


r346=0


r347=0


r348=0


r349=0


r350=0


r351=0


r352=0


       - a22*r328 + b12*kap*r315
r353=----------------------------
                 b12


r354=0


r355=0


r40=0


r41=0


r42=0


r43=0


r44=0


r45=0


r46=0


r47=0


r48=0


r49=0


r410=0


r411=0


r412=0


r413=0


r415=0


r416=0


r417=0


r418=0


r419=0


r420=0


r421=0


r423=0


r424=0


r426=0


r427= - r4110


r428=0


r429=0


r430= - r4110


r431=r488


r432=0


r433=0


r435=0


r436=0


r437=0


r438=0


r439=0


r440=0


r441=0


r442=0


r443=0


r444=0


r445=0


r446=2*r4110


r447=0


r448=0


r449=0


r450=0


r451=0


r452=r488


r453= - r487


r454=0


r455=0


r456=0


r457=r4110


r458=0


r459=0


r460=0


r461=0


r462=0


r463=0


      a22*r487 - b12*kap*r4110
r464=--------------------------
                b12


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


r479=0


r480=0


r482=0


r483=2*r4110


r484=0


r485=0


r486=0


r489=0


r490=0


r491=0


r492=0


r493=0


r494=2*r4110


r495=0


r496=0


r497=0


r498=0


r499=0


r4100=0


r4101=0


r4102=0


r4103=0


r4104=0


r4105=0


r4106=0


r4108=0


r4109=0


r4111=0


r4112=0


r4113=0


       a22*r487 - b12*kap*r4110
r4114=--------------------------
                 b12


r4115=0


r4116=0


r4117=0


r4118=0


r4119=0


r4120=0


r4121=0


r4122=0


r4123=0


r4124=0


r4125=0


m2=0


m1=0


n2=0


n1=0


a33=0


a23=0


a13=0


a11=a22

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:

 r13, r29, r315, r319, r328, r343, r434, r487, r4110, r488, 
m3, n3, a22, b12

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.

 
{{m3,n3},a22,b12}

Relevance for the application:

Modulo the following equation:

     2      2
0=a22  + b12 *kap


the system of equations related to the Hamiltonian HAM:

      2                     2
HAM=u1 *a22 + u1*v2*b12 + u2 *a22 - u2*v1*b12 + u3*n3 + v3*m3

has apart from the Hamiltonian and Casimirs the following 10 first integrals: 

        2           2        3
FI=u1*u3 *v1 + u2*u3 *v2 + u3 *v3

  = a product of the elements of: {u3,

   u3,

   u1*v1 + u2*v2 + u3*v3}

{HAM,FI} = 0



        2   2         2   2                                     2   2
FI= - u1 *u3 *kap + u1 *v1  + 2*u1*u2*v1*v2 + 2*u1*u3*v1*v3 - u2 *u3 *kap

        2   2                     2   2     2   2
    + u2 *v2  + 2*u2*u3*v2*v3 - u3 *v1  - u3 *v2

which the program can not factorize further.

{HAM,FI} = 0



     2   2            2            2   2            2            2
FI=u1 *u3 *a22 + u1*u3 *v2*b12 + u2 *u3 *a22 - u2*u3 *v1*b12 + u3 *v3*m3

  = a product of the elements of: {u3,

   u3,

     2                     2
   u1 *a22 + u1*v2*b12 + u2 *a22 - u2*v1*b12 + v3*m3}

{HAM,FI} = 0



     4
FI=u3

  = a product of the elements of: {u3,u3,u3,u3}

{HAM,FI} = 0



                           2
FI=u1*u3*v1 + u2*u3*v2 + u3 *v3

  = a product of the elements of: {u3,u1*v1 + u2*v2 + u3*v3}

{HAM,FI} = 0



        2                                      2
FI= - u1 *u3*a22 - u1*u3*v2*b12 + u1*v1*m3 - u2 *u3*a22 + u2*u3*v1*b12

    + u2*v2*m3

which the program can not factorize further.

{HAM,FI} = 0



     3
FI=u3

  = a product of the elements of: {u3,u3,u3}

{HAM,FI} = 0



     2            2               2        2        2
FI=u1 *u3*kap + u2 *u3*kap + u3*v1  + u3*v2  + u3*v3

                                        2         2         2     2     2
  = a product of the elements of: {u3,u1 *kap + u2 *kap + v1  + v2  + v3 }

{HAM,FI} = 0



     2
FI=u3

  = a product of the elements of: {u3,u3}

{HAM,FI} = 0



FI=u3

which the program can not factorize further.

{HAM,FI} = 0





And again in machine readable form:



HAM=u1**2*a22 + u1*v2*b12 + u2**2*a22 - u2*v1*b12 + u3*n3 + v3*m3$

FI=u1*u3**2*v1 + u2*u3**2*v2 + u3**3*v3$

FI= - u1**2*u3**2*kap + u1**2*v1**2 + 2*u1*u2*v1*v2 + 2*u1*u3*v1*v3 - u2**2*u3**
2*kap + u2**2*v2**2 + 2*u2*u3*v2*v3 - u3**2*v1**2 - u3**2*v2**2$

FI=u1**2*u3**2*a22 + u1*u3**2*v2*b12 + u2**2*u3**2*a22 - u2*u3**2*v1*b12 + u3**2
*v3*m3$

FI=u3**4$

FI=u1*u3*v1 + u2*u3*v2 + u3**2*v3$

FI= - u1**2*u3*a22 - u1*u3*v2*b12 + u1*v1*m3 - u2**2*u3*a22 + u2*u3*v1*b12 + u2*
v2*m3$

FI=u3**3$

FI=u1**2*u3*kap + u2**2*u3*kap + u3*v1**2 + u3*v2**2 + u3*v3**2$

FI=u3**2$

FI=u3$