Two unit- speed searchers at (0,0) seek a randomly located target on the plane accoirding to a known unsymmetric continous distribution. The objective is to minimize the expected time for the searchers to return to (0,0) after one of them has found the target. We find a necessary conditions which make the search strategy be optimal when the target has a bivariate Balakrishnan skew--normal distribution. The search strategy is derived using a dynamic programming algorithm. An example is given to show the applications of this technique. The problem has applications to parallel processing and to the optimal choice of drilling depths in the search for an underground mineral.