  We consider a pair of coupled non-linear partial differential equations
describing a biochemical model system. The Weiss-algorithm for the Painle\'{e}
test, that has been succesfully used in mathematical physics for the
KdV-equation, Burgers equation, the sine-Gordon equation etc., is applied, and
we find that the system possesses only the "conditional" Painlev\'{e} property.
We use the outcome of the analysis to construct an auto-B\"{a}cklund
transformation, and we find a variety of one and two-parameter families of
special solutions.
