Integrability of Hamiltonian systems with gyroscopic term

Abstract We study the integrability of 2D Hamiltonian systems $$H_\mu =\frac{1}{2}(p_1^2+p_2^2) + \omega (p_1q_2-p_2q_1) -\tfrac{\mu }{r}+ V(q_1,q_2)$$ H μ = 1 2 ( p + ) ω q - r V , , where $$r^2=q_1^2+q_2^2$$ and potential $$V(q_1,q_2)$$ is a homogeneous rational function integer degree k . The main result states that under very general assumptions: $$\mu \ne 0$$ ≠ 0 $$|k|>2$$ | k > $$V(1,\mathrm {i})\ne i or $$V(-1,\mathrm system not integrable. It was obtained by combining Levi-Civita regularization, differential Galois methods, so-called coupling constant metamorphosis transformation. proof regularized version problem integrable contains most important theoretical results, which can be applied to other problems.