We study the integrability of a two-dimensional Hamiltonian system with a gyroscopic term and a non-homogeneous potential composed of two homogeneous components of different degrees. The model describes the motion of a particle in a plane under the combined influence of a central (Kepler-type) potential, a uniform magnetic field, and a superposition of homogeneous forces. By combining the Levi–Civita regularization with the so-called coupling constant metamorphosis transformation, and employing differential Galois theory, we derive analytical necessary conditions for integrability in the Liouville sense. They put restrictions on the degrees of homogeneity of the potential terms and their values in particular points. The obtained results encompass and generalize several classical galactic and astrophysical models, including the generalized Hill model, the Hénon–Heiles and Armbruster–Guckenheimer–Kim systems, providing a unified framework for studying non-homogeneous Hamiltonians. We demonstrate the effectiveness of the derived integrability obstructions by proving the non-integrability of these models in the presence of a uniform rotational field. The numerical analysis via the Poincaré cross-sections further confirms the analytical results, illustrating the transition from regular to chaotic dynamics as the rotational and non-homogeneous terms are introduced. Moreover, we show that, without the Kepler-type term, a generalized non-homogeneous extension of the exceptional potential remains integrable. The explicit forms of the first integrals are given.
Szumiński et al. (Sun,) studied this question.