  It is shown that the regular-at-infinity solution of the 1D Schrodinger
equation with the hyperbolic Poschl-Teller (h-PT) potential with integer
parameters is expressible in terms of a n-order Heun polynomial in y=thr at an
arbitrary negative energy. It was proven that the Heun polynomials in question
form a subset of generally complex Lambe-Ward polynomials corresponding to zero
value of the accessory parameter. Since the mentioned solution expressed in the
new variable y has an almost-everywhere holomorphic (AEH) form it can be used
as the factorization function (FF) for canonical Liouville-Darboux
transformations (CLDTs) to construct a continuous family of shape-invariant
rational potentials exactly-solvable by the Hp-seed (HpS) Heine polynomials.
There are also two sequences of infinitely many rational potentials generated
using CLDTs with nodeless regular-at-origin AEH FFs.
