A systematic procedure to study one-dimensional Schroedinger equation with position-dependent effective mass (PDEM) in the kinetic energy operator is explored. We show that any constant mass Schroedinger equation can be coordinate transformed in terms of some mass function such that the resulting form matches with the PDEM equation and the ambiguity parameters are decoupled in the potential term. This enables us to derive many exact results including those related to isospectrality. We also seek interesting connections with the so(2,1) algebra by choosing appropriate forms of the underlying generators.