Integrability of the Yang-Mills Hamiltonian system

This paper considers the integrability of generalized Yang-Mills system with the HamiltonianH_a(p, q)=1/2(p₁²+p₂²+a₁q₁²+a₂q₂²)+1/4q₁⁴+1/4a₃q₂⁴+ 1/2a₄q₁²q₂². We prove that the system is integrable for the cases: (A)a₁=a₂,a₃=a₄=1; (b)a₁=a₂,a₃=1,a₄=3; (C)a₁=a₂/4,a₃=16,a₄=6. Our main result is the presentation of these integrals. Only for cases A and B does the Yang-Mills Hamiltonian possess the Painlevé property. Therefore the Painlevé test does not take account of the integrability for the case C.