A constructive algorithm is presented for solving a two-dimensional (2-D) polynomial matrix equation that plays an important role in the output feedback stabilization of multivariable 2-D linear systems. It is shown that this algorithm can have some computational advantages over the previous known ones based on the Gröbner-basis approach. As an application of the algorithm, the class of all stabilizing strictly causal 2-D compensators is parametrized for a given stabilizable causal (not necessarily strictly causal) 2-D plant.