  We study the P"oschl-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 a interesting
interpretation in terms of finite lattice spin (L-2)/2 XXZ quantum chain with
Delta= cos(pi/L) (for free-free boundary conditions), or Delta=-cos(pi/L) (for
periodic boundary conditions). This result generalises 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.
