  Systems of differential equations with polynomial right-hand sides are very
common in applications. In particular, when restricted to the positive orthant,
they appear naturally (according to the law of mass-action kinetics) in
ecology, population dynamics, as models of biochemical interaction networks,
and models of the spread of infectious diseases. On the other hand, their
mathematical analysis is very challenging in general; in particular, it is very
difficult to answer questions about the long-term dynamics of the variables
(species) in the model, such as questions about persistence and extinction.
Even if we restrict our attention to mass-action systems, these questions still
remain challenging. On the other hand, if a polynomial dynamical system has a
weakly reversible single linkage class ($W\!R^1$) realization, then its
long-term dynamics is known to be remarkably robust: all the variables are
persistent (i.e., no species goes extinct), irrespective of the values of the
parameters in the model. Here we describe an algorithm for finding $W\!R^1$
realizations of polynomial dynamical systems, whenever such realizations exist.
