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In this paper, we examine a generalized magma equation for rational values oftwo parameters, $m$ and $n$. Firstly, the similarity reductions are found usingthe Lie group method of infinitesimal transformations. The Painlev\'e ODE testis then applied to the travelling wave reduction, and the pairs of $m$ and $n$which pass the test are identified. These particular pairs are furthersubjected to the ODE test on their other symmetry reductions. Only two casesremain which pass the ODE test for all such symmetry reductions and these arecompletely integrable. The case when $m=0$, $n=-1$ is related to theHirota-Satsuma equation and for $m=\tfrac12$, $n=-\tfrac12$, it is a real,generalized, pumped Maxwell-Bloch equation.

Painlevé Analysis and Similarity Reductions for the Magma Equation