  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 KdV equations and three variants of generalized
MKdV equations. 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 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 sytems with variable
coefficients. Hence we attempt to apply the Estabrook-Wahlquist (EW)
prolongation technique, a relatively self-consistent procedure requiring little
prior information. However, this immediately requires that the technique be
significantly generalized or broadened in several different ways, including
solving matrix partial differential equations instead of algebraic ones. The
new and extended EW technique whch results is illustrated by algorithmically
deriving generalized Lax-integrable versions of the generalized fifth-order
KdV, and MKdV equations.
