Integrability of certain Hamiltonian systems in $2D$ variable curvature spaces

The objective of this work is to examine the integrability of Hamiltonian systems in $2D$ spaces with variable curvature of certain types. Based on the differential Galois theory, we announce the necessary conditions of the integrability. They are given in terms of arithmetic restrictions on values of the parameters describing the system. We apply the obtained results to some examples to illustrate that the applicability of the obtained result is easy and effective. Certain new integrable examples are given. The findings highlight the applicability of the differential Galois approach in studying the integrability of Hamiltonian systems in curved spaces, expanding our understanding of nonlinear dynamics and its potential applications.