Finite lattice Bethe ansatz systems and the Heun equation

We study the Poschl-Teller equation in complex domain and deduce infinite families of TQ and Bethe ansatz equations, classified by four integers. In all these models the form of T is very simple, while Q can be explicitly written in terms of the Heun function. At particular values there is an interesting interpretation in terms of finite lattice spin-L-2/2 XXZ quantum chain with Delta = cospi/L (for free-free boundary conditions), or Delta = -cospi/L (for periodic boundary conditions). This result generalizes the findings of Fridkin, Stroganov and Zagier. We also discuss the continuous (field theory) limit of these systems in view of the so-called ODE/IM correspondence.