Two-dimensional integrable systems with position-dependent mass via complex holomorphic functions

We study the relationship between integrable systems with a position-dependent mass (PDM) and complex holomorphic functions and the potential applications of the latter to deduce the former. For a prescribed mass term the associated complex function is derived. The complex function and related plane transformation are used to generate the PDM systems of three integrable Henon-Heiles systems and a Holt system as well. We also figure out a holomorphic function, which ensures separability of the corresponding PDM systems in the polar-like coordinates. The holomorphic function together with Jacobi method have yielded a variety of generalized separable systems. At last we put forward an example of a family of separable systems to show that not all PDM systems can be deduced through some holomorphic function.