  Detailed mappings between zero-curvture equations for prolongation structures
of nonlinear pde's and Estabrook-Wahlquist algorithms for same are given. The
differences are exemplified by studies of the sine-Gordon equation. An example
where the prolongation structure must be infinite-dimensional is given by the
Robinson-Trautman equation, where the minimal algebra is $K_2$. In general
these algorithms require integration of vector-field valued pde's; solutions of
simultaneous flow equations are given. Applications to coupled systems of flow
equations are given, where the result describes Lie algebras of vector fields
vertical over fibers of pseudopotentials over a jet bundle appropriate for a
given system of pde's; algebras invariant under sl(2,C) are of special
interest.
