Hamiltonians with so3^2 and partial first degree integrals

I. Hamiltonians admitting partial integrals with factor U3,
I.A. Hogeneous case

I.A.a) First integrals with factor U3
degree 1: none
degree 2: s1, s2, s3, s4, s5
degree 3: s1, s2, s3, s4, s5
degree 4: s1, s2, s3, s4, s5, s6, s7, s8, s9
degree 5: s1, s2, s3, s4, s5
degree 6: s1, s2, s3, s4, s5, s6, s7

I.A.b) General frst integrals
degree 1: s1, s2, s3, s4, s5
degree 2: s1, s2, s3, s4, s5, s6
degree 3: s1, s2, s3, s4, s5, s6
degree 4: s1, s2, s3, s4, s5, s6, s7, s8
degree 5: s1, s2, s3, s4, s5
degree 6: s1, s2, s3, s4, s5, s6, s7, s8

I.B. Inhogeneous case

I.B.a) First integrals with factor U3
degree 1: none
degree 2: s1, s2
degree 3: s1, s2, s3, s4, s5
degree 4: s1, s2, s3, s4, s5, s6, s7, s8, s9, s10, s11, s12
degree 5:
degree 6:

I.B.b) General frst integrals
degree 1: s1, s2, s3, s4, s5
degree 2: s1, s2, s3, s4, s5, s6
degree 3: s1, s2, s3, s4, s5, s6, s7, s8
degree 4: s1, s2, s3, s4, s5, s6, s7, s8, s9, s10, s11, s12, s13
degree 5:
degree 6:

II. Hamiltonians admitting partial integrals with factor U3+V3,
II.A. Homogeneous case

II.A.a) First integrals with factor U3+V3
degree 1: none
degree 2: s1, s2, s3, s4, s5
degree 3: s1
degree 4:
degree 5:
degree 6:

II.A.b) General frst integrals
degree 1: s1
degree 2: s1, s2, s3, s4, s5, s6
degree 3: s1, s2, s3
degree 4:
degree 5:
degree 6:

II.B. Inhomogeneous case

II.B.a) First integrals with factor U3+V3
degree 1: none
degree 2: s1, s2, s3, s4, s5
degree 3: s1, s2
degree 4: running
degree 5:
degree 6: running

II.B.b) General frst integrals
degree 1: s1
degree 2: s1, s2, s3, s4, s5, s6, s7
degree 3: s1, s2, s3
degree 4: running
degree 5:
degree 6: running

III. Hamiltonians admitting partial integrals with factor aU3+bV3,
III.A. Homogeneous case

III.A.a) First integrals with factor aU3+bV3
degree 1: none
degree 2: none
degree 3: none
degree 4: none
degree 5: none
degree 6: none

III.A.b) General frst integrals
degree 1: s1
degree 2: s1
degree 3: s1
degree 4: s1, s2
degree 5: s1, s2
degree 6: s1

III.B. Inhomogeneous case

III.B.a) First integrals with factor aU3+bV3
degree 1: none
degree 2: none
degree 3: none
degree 4: none
degree 5: none
degree 6: none

III.B.b) General frst integrals
degree 1: s1
degree 2: s1
degree 3: s1, s2
degree 4: s1
degree 5: s1
degree 6: