  The paper re-examines Milson's analysis of the rational Sturm-Liouville (RSL)
problem with two complex conjugated regular singular points -i and +i by taking
advantage of Stevenson's complex linear-fraction transformation S(y) of the
variable y restricted to the real axis. It was explicitly demonstrated that
Stevenson's hypergeometric polynomials in a complex argument S are nothing but
Romanovsky polynomials converted from y to S. The use of Stevenson's
mathematical arguments unambiguously confirmed 'exact solvability' of the
Milson potential. It was revealed that the Milson potential has two branches
referred to as 'inside' and 'outside' depending on positions of zeros of the
so-called 'tangent polynomial' (TP) relative to the unit circle. The two
intersect along the shape-invariant Gendenshtein (Scarf II) potential. The
remarkable feature of the RCSLE associated with the inner branch of the Milson
potential (as well as its shape-invariant limit) is that it has two sequences
of nodeless almost-everywhere holomorphic (AEH) solutions which can be used as
factorization functions (FFs) for constructing new quantized-by-polynomials
potentials. In case of the Gendenshtein potential complex-conjugated
characteristic exponents (ChExps) at finite singular points of the given RCSLE
become energy independent so that each polynomial sequence turns into a finite
set of orthogonal polynomials. This confirms Quesne's conjecture [J. Math.
Phys. 54 122103 (2013)] that the 'Case III' polynomials discovered by her can
be used for constructing orthogonal polynomials of novel type.
