A piecewise deterministic Markov process with random switching between flows, occuring e.g. in certain stochastic models for gene expression, is considered in this article. The main goal is to provide a set of reasonable conditions under which any stationary distribution of the process that corresponds to an ergodic stationary distribution of the discrete-time Markov chain given by the post-jump locations of this process has a density with respect to the Lebesgue measure.