  Equilibrium measures in the real axis in the presence of rational external
fields are considered. These external fields are called rational since their
derivatives are rational functions. We analyze the evolution of the equilibrium
measure, and its support, when the size of the measure, $t$, or other
parameters in the external field vary. Our analysis is illustrated by studying
with detail the case of a generalized Gauss-Penner model, which, in addition to
its mathematical relevance, has important physical applications (in the
framework of random matrix models). This paper is a natural continuation of
\cite{MOR2013}, where equilibrium measures in the presence of polynomial
external fields are thoroughly studied.
