  Phylogenetic varieties related to equivariant substitution models have been
studied largely in the last years. One of the main objectives has been finding
a set of generators of the ideal of these varieties, but this has not yet been
achieved in some cases (for example, for the general Markov model this involves
the open "salmon conjecture") and it is not clear how to use all generators in
practice. Motivated by applications in biology, we tackle the problem from
another point of view. The elements of the ideal that could be useful for
applications in phylogenetics only need to describe the variety around certain
"points of no evolution". We produce a collection of explicit equations that
describe the variety on a Zariski open neighborhood of these points. Namely,
for any tree on any number of leaves (and any degrees at the interior nodes)
and for any equivariant model on any set of states, we compute the codimension
of the corresponding phylogenetic variety. We prove that this variety is smooth
at general points of no evolution, and provide an algorithm to produce a
complete intersection that describes the variety around these points.
