  Variable Coefficient Korteweg de Vries (vcKdV), Modified Korteweg de Vries
(vcMKdV), and nonlinear Schrodinger (NLS) equations have a long history dating
from their derivation in various applications. A technique based on extended
Lax Pairs has been devised recently to derive variable-coefficient
generalizations of various Lax-integrable NLPDE hierarchies. The resulting Lax-
or S-integrable NLPDEs have both time- AND space-dependent coefficients, and
are thus more general than almost all cases considered earlier via other
methods such as the Painlev\'e 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 generalization 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 embark in
this paper on an attempt to systematize the derivation of Lax-integrable
systems with variable coefficients. An ideal approach would be a method which
does not require knowledge of the Lax pair to an associated constant
coefficient system, and also involves little to no guesswork. Hence we attempt
to apply the Estabrook-Wahlquist (EW) prolongation technique, a relatively
self-consistent procedure requiring little prior information.
  The new and extended EW technique which results is illustrated by
algorithmically deriving generalized Lax-integrable versions of the NLS,
generalized fifth-order KdV, MKdV, and derivative nonlinear Schrodinger (DNLS)
equations.
