  A nonlinear coupled system descriptive of multi-ion electrodiffusion is
investigated and all parameters for which the system admits a single-valued
general solution are isolated. This is achieved \textit{via} a method initiated
by Painleve' with the application of a test due to Kowalevski and Gambier. The
solutions can be obtained explicitly in terms of Painleve' transcendents or
elliptic functions.
