  In this paper we continue the study of good re-embeddings of affine
K-algebras started in [KLR]. The idea is to use special linear projections to
find isomorphisms between a given affine K-algebra K[X]/I, where
X=(x_1,...,x_n), and K-algebras having fewer generators. These projections are
induced by particular tuples of indeterminates Z and by term orderings $\sigma$
which realize Z as leading terms of a tuple F of polynomials in I. In order to
efficiently find such tuples, we provide two major new tools: an algorithm
which reduces the check whether a given tuple F is Z-separating to an LP
feasibility problem, and an isomorphism between the part of the Gr\"obner fan
of I consisting of marked reduced Gr\"obner bases which contain a Z-separating
tuple and the Gr\"obner fan of the intersection of I and K[X\Z]. We also
indicate a possible generalization to tuples Z which consist of terms. All
results are illustrated by explicit examples.
