  We present a polyhedral algorithm to manipulate positive dimensional solution
sets. Using facet normals to Newton polytopes as pretropisms, we focus on the
first two terms of a Puiseux series expansion. The leading powers of the series
are computed via the tropical prevariety. This polyhedral algorithm is well
suited for exploitation of symmetry, when it arises in systems of polynomials.
Initial form systems with pretropisms in the same group orbit are solved only
once, allowing for a systematic filtration of redundant data. Computations with
cddlib, Gfan, PHCpack, and Sage are illustrated on cyclic $n$-roots polynomial
systems.
