Skew-symmetric Hamiltonians of so(4) and so(3,1) with kappa

The ansatz:

 STRUC_CONS:={{U1,U2, U3}, {U2,U3, U1}, {U3,U1, U2},
              {U1,V2, V3}, {U2,V3, V1}, {U3,V1, V2},
              {U1,V3,-V2}, {U2,V1,-V3}, {U3,V2,-V1},
              {V1,V2, kap*U3}, {V2,V3, kap*U1}, {V3,V1, kap*U2}}$

 HAM := n1*U1 + n2*U2 + n3*U3 + 
	m1*V1 + m2*V2 + m3*V3 + 
	  a11*U1**2 +   a22*U2**2 +   a33*U3**2 + 
	2*a12*U1*U2 + 2*a23*U2*U3 + 2*a13*U1*U3 +
	  b11*U1*V1 +   b12*U1*V2 +   b13*U1*V3 + 
	  b21*U2*V1 +   b22*U2*V2 +   b23*U2*V3 + 
	  b31*U3*V1 +   b32*U3*V2 +   b33*U3*V3 + 
	  c11*V1**2 +   c22*V2**2 +   c33*V3**2 +
	2*c12*V1*V2 + 2*c23*V2*V3 + 2*c13*V1*V3 $

 J1 := kap*(U1**2+U2**2+U3**2) + V1**2+V2**2+V3**2
 J2 := U1*V1 + U2*V2 + U3*V3

c11:=c22:=c33:=c12:=c23:=c13:=0$

b11:=b22:=b33:=0$  b21:=-b12$  b31:=-b13$  b32:=-b23$

a12:=0$ b13:=0$ b23:=0$

b12 <> 0

U1*V2, V1**2, U3*V3

m1:=m2:=m3:=n1:=n2:=n3:=0

Only nontrivial homogeneous solutions are computed, for example, not the 1st degree solution again for all other degrees. In other words, when computing the tail for the 3rd, 4th and 6th degree Hamiltonians, it was assumed that one of the 3 inequalities had to be satisfied:
a11-a22<>0, or a13<>0 or a23<>0.

When computing the 4th and 6th degree tails it was in addition assumed that:
a11-a22<>0, or a11+a33<>0, or a11^2+a13^2+a23^2+b12^2*kap <> 0,
and so on. The prize is that, for example, for the 1st degree Hamiltonian it is not known whether for subcases of it there are additional tails with inhomogeneous first integrals. These could be computed if they are of interest.

For the mystic case

 c2:= 1$ c1:= 1$ a11:=a22:=c1*a3$
a33:=(c1+c2)*a3$ a13:=c2*a1/2$ a23:=0$ b12:=1$ kap:=-(a1**2+a3**2)$

What we will call deformations is to take for the quadratic part of the Hamiltonian that one which has a first degree first integral, i.e. we set 0=a11-a22=a13=a23 with an arbitrary linear tail (arbitrary m1,m2,m3,n1,n2,n3) and arbitrary inhomogeneous first integrals of degree 2,3,.. .

First Integrals of degree 1

First Integrals of degree 2

J2 <> 0
Homogeneous solution(s): none
Their inhomogeneous generalization(s): none
Deformations: has only first degree first integral
The general inhomogeneous solution(s): none

J2 = 0
Homogeneous solution(s): none
Their inhomogeneous generalization(s):
The general inhomogeneous solution(s):

First Integrals of degree 3

J2 <> 0
Homogeneous solution(s): 1
Their inhomogeneous generalization(s): 1.1
Deformations: 1, 2,
The general inhomogeneous solution(s): same as 1.1 (only with gauge a13=0).

J2 = 0
Homogeneous solution(s): 1 (same as homog. for J2<>0), 2(seems new)
Their inhomogeneous generalization(s): 1 (same as for inhomog. for J2<>0)
The general inhomogeneous solution(s):

First Integrals of degree 4

J2 <> 0
Homogeneous solution(s): 1, 2, 3 (symmetric to 2), 4, 5 (super-integrable)
Vector form of Homogeneous solution(s): case (2.6), case (2.7)
Their inhomogeneous generalization(s): These are re-computed and should be ok: 1.1, 2.1, 3.1 (symmetric to 2.1), 4.1
Deformations: 1, 2, 3, 4
The general inhomogeneous solution(s): same as 1.1, 2.1, 3.1, 4.1

J2 = 0
Homogeneous solution(s):
    1 (a special case (intersection) of homog. 1,2,3 for J2<>0),
    2 (same as homog. 1 for J2<>0),
    3 (same as homog. 3 for J2<>0),
    4, (same as homog. 4 for J2<>0),
    5 (seems new),
    6, (seems new),
    7 (same as homog. 2 for J2<>0),
Their inhomogeneous generalization(s):
     1.1 same as 1.1 for J2<>0, 1.2 special case of 1.1 for a23=0, 1.3 special case of 1.1 for n3=0, 1.4 special case of 1.1 for a11=n2=n3=0
    2.1same as for J2<>0
    3.1same as for J2<>0
    4.1same as for J2<>0
Deformations: 1
The general inhomogeneous solution(s):

First Integrals of degree 5

J2 <> 0
Homogeneous solution(s): none
Their inhomogeneous generalization(s): none
Deformations: 1
The general inhomogeneous solution(s): These are all special cases of the first, third and fourth degree first integral solutions shown above.

J2 = 0
Homogeneous solution(s): none
Their inhomogeneous generalization(s): no solutions --> no generalizations
The general inhomogeneous solution(s):

First Integrals of degree 6

J2 <> 0
Homogeneous solution(s): 1,
Their inhomogeneous generalization(s): For these solutions we have a special setting:

a11=a22=a3, a33=a3/2, a13=-a1/4, a23=-a2/4, kap=-(a1^2+a2^2+a3^2).

J2 = 0
Homogeneous solution(s):
    1 (special case of 6th degree J2<>0),
    2 (=3rd degree J2<>0),
    3 (new with a11=a22=3*a33),
    4 (special case of 3rd degree, J2<>0),
    5 (special case of 3rd degree, J2<>0),
    6 (=sol 1 of 4th degree, J2<>0),
    7 (=sol 4 of 4th degree, J2<>0),
    8 (=sol 2 of 4th degree, J2<>0),
    9 (special case of sol 2 of 4th degree, J2<>0),
    10 (=1st degree J2<>0),
    11 (=sol 2 of 4th degree, J2<>0),
    12 (special case of sol 3 of 4th degree, J2<>0),
    13 (=6th degree J2<>0)
Their inhomogeneous generalization(s):
    The only true homogeneous 6th degree, J2<>0 solution (number 1 above) generalized with linear tail under J2=0:
         1 (generalizes 1.1 above through m3 being arbitrary now and by that includes also 1.2 and 1.3),
         2 (special case of the first of these 3),
         3 (special case of the first of these 3)
The general inhomogeneous solution(s):

First Integrals of degree 7

J2 <> 0
Homogeneous solution(s): none
Their inhomogeneous generalization(s): none
The general inhomogeneous solution(s): ?

J2 = 0
Homogeneous solution(s): lack of memory, dies in case 3.2. see skew/i7/homo/J2.log
Their inhomogeneous generalization(s): already conditions too large
The general inhomogeneous solution(s):

First Integrals of degree 8

J2 <> 0
Homogeneous solution(s): All special solutions that were obtained in attempts from December 2003 are contained above.
Their inhomogeneous generalization(s): ?
The general inhomogeneous solution(s): ?

J2 = 0
Homogeneous solution(s): already conditions too large,
Their inhomogeneous generalization(s):
The general inhomogeneous solution(s):