Abstract: The nonlinear Schrödinger (NLS) equation is an integrable universal model that describes the evolution of slowly varying envelope of a quasi-monochromatic wave in weakly nonlinear media. From both mathematical and physical point of view, the dynamics of self-focusing media governed by the focusing NLS equation with periodic boundary conditions is a classical research topic, which is associated with a non-self-adjoint Dirac operator with periodic potentials. In this talk, we present a novel, explicit two-parameter family of finite-band Jacobi elliptic potentials for a non-self-adjoint Dirac operator, which connects two previously known limiting cases in which the elliptic parameter is zero or one. A full characterization of the spectrum and the connection problems for Heun’s equation are discussed.