  The paper presents the unified technique for constructing SUSY ladders of
rational Liouville potentials (RLPs) starting from the so-called
"Gauss-reference" (GRef) potentials exactly quantized on the line via classical
Jacobi, classical (generalized) Laguerre, or Romanovski-Routh polynomials with
energy-dependent indexes. Each RLP is obtained by means of the Liouville
transformation (LT) of the appropriate rational canonical Sturm-Liouville
equation (RCSLE) with second-order poles. The presented analysis takes
advantage of the generic factorization of canonical Sturm-Liouville equations
(CSLEs) in terms of intertwining "generalized" Darboux operators. We refer to
the latter operators as the canonical Liouville-Darboux transformations (CLDTs)
to stress that they are equivalent to three-step operations: i) the LT from the
CSLE to the Schrodinger equation; ii) the Darboux transformation (DT) of the
appropriate LP; and iii) the inverse LT from the Schrodinger equation to the
new CSLE. It is proven that the CLDT preserves the rational form of the RCSLE
if its factorization function (FF) is an almost-everywhere holomorphic (AEH)
solution of the RCSLE (or, in other words, a solution with a rational
logarithmic derivative). As explained in the paper there are up to four gauge
transformations which convert each RCSLE of our interest into the second-order
differential equations with energy-dependent polynomial coefficients. The most
important result of the paper is that polynomial solutions of these equations
belong to sequences of Heine polynomials obtained by varying free terms at
fixed values of singular points and the appropriate characteristic exponents.
This allows us to construct networks of polynomial solutions -- the so-called
"r-, c-, or i-Gauss-seed" (r-, c-, or i-GS) Heine polynomials -- starting from
Jacobi, (generalized) Laguerre or Routh polynomials, respectively.
