This paper finds search plans, which are optimal in the sense of minimizing mean time to find the target, for several types of searches for a stationary target, in the presence of Poisson-distributed false targets. One must allocate the available effort between broad search to develop contacts and contact investigation to ascertain whether these contacts are real or false targets; one must further allocate within each of these two types of effort. A search plan is described by a pair of functions $(m,\lambda )$, where m gives the broad search effort density as a function of broad search time and location, and $\lambda $ gives the amount of contact investigation effort that one is willing to exert in attempting to identify a contact as a function of the location of the contact and broad search time. Two general optimization theorems presented in this paper are applied to find optimal search plans. Both theorems may be thought of as extensions of a Neyman-Pearson technique to allocations of more than one ty...