The path planning problem for a controlled moving object with a nonuniform radiation pattern is considered for the case in which the necessary path optimality conditions are degenerate. Additional constraints are introduced, and two optimization problems are formalized, one of finding an optimal speed mode for a given path of the object and the other of finding an optimal path when moving at a constant speed. Methods and algorithms for constructing optimal paths and finding speed modes are proposed; for the second problem, the analytical domain of existence of a solution for arbitrary parameters is found. The analytical results are illustrated with examples.