  In this paper we study the regularity of weak solutions to an
elliptic-parabolic system modeling natural network formation. The system is
singular and involves cubic nonlinearity. Our investigation reveals that weak
solutions are H\"{o}lder continuous when the space dimension $N$ is $2$. This
is achieved via an inequality associated with the Stummel-Kato class of
functions and refinement of a lemma originally due to S. Campanato and C. B.
Morrey (\cite{G}, p. 86).
