Abstract
•A new theorem is introduced to study the integrability of a Hamiltonian system with certain type of nonhomogeneous potential.•This new theory is more effective in studying the integrability of galactic potentials systematically.•A new integrable problem which generalizes the Swinging Atwood machine is announced.
In this work, we inspect the integrability of a natural Hamiltonian system interpreted physically as the motion of a particle in the Euclidean plane under the effect of conservative forces derived from a certain type of a non-homogeneous potential. We announce the necessary conditions for its integrability by using the differential Galois theorem. We present three examples to clarify the applicability of the obtained results is easy and efficacious. Some of these examples restore the previous results in the literature, and one of them gives a new integrable case describing a generalization of the well-known Swinging Atwood machine.