  This paper develops two approaches to Lax-integrbale systems with
spatiotemporally varying coefficients. A technique based on extended Lax Pairs
is first considered to derive variable-coefficient generalizations of various
Lax-integrable NLPDE hierarchies recently introduced in the literature. As
illustrative examples, we consider generalizations of the NLS and DNLS
equations, as well as a PT-symmetric version of the NLS equation. It is
demonstrated that the techniques yield Lax- or S-integrable NLPDEs with both
time- AND space-dependent coefficients which are thus more general than almost
all cases considered earlier via other methods such as the Painleve Test, Bell
Polynomials, and various similarity methods. However, this technique, although
operationally effective, has the significant disadvantage that, for any
integrable system with spatiotemporally varying coefficients, one must 'guess'
a generaliza- tion of the structure of the known Lax Pair for the corresponding
system with constant coefficients. Motivated by the somewhat arbitrary nature
of the above procedure, we therefore next attempt to systematize the derivation
of Lax-integrable sytems with variable coefficients. We attempt to apply the
Estabrook- Wahlquist (EW) prolongation technique, a relatively self-consistent
procedure requiring little prior infomation. However, this immediately requires
that the technique be significantly generalized or broadened in several
different ways. The new and extended EW technique which results is illustrated
by algorithmically deriving generalized Lax-integrable versions of NLS,
PT-symmetric NLS, and DNLS equations.
