A method of evasion for linear differential games was established by Pontryagin in his work [6]. He considered there an evasion game governed by a linear differential equation in R”, with a linear subspace M of R” as a terminal set. The game was assumed to end when the solution to the equation reached coincidence with the subspace M. The evader’s aim was to prevent the termination of the game. The method of evasion applied in [6] was based on the projection of the trajectory onto suitably chosen two-dimensional subspace L laying in the orthogonal complement of M in R” and, thereby, reduced the evasion problem to the problem of avoidance of the origin in the subspace L. (It was assumed, as usually in such a case, that dim M< IZ 2.) Next, this method has been extended by many authors to the nonlinear case, see, e.g., c4, 7, 31. When more than one pursuer takes part in the game, some additional assumptions (like dim M< n 3, as it was made in [3]) are necessary in order that Pontryagin’s method be applicable to such an evasion problem. There are, however, examples of games with several pursuers that do not satisfy these extra assumptions but, nevertheless, can be solved directly, see [2, 10, 11, 91. In this paper, we treat the problem of evasion for games with many pursuers governed by kth order differential equations on the plane using a method closely related to the one given in [9]. This problem resembles the above-mentioned general evasion problems in spirit, and, simultaneously, allows us to incorporate many special examples. We limit ourselves here to considering the case of games described by kth order differential equations instead of more general situations, in which the operator V (see [7]) might be used, as the expense of being able to assert the possibility of “evasion along each trajectory” in these games. The assumption that the players move on the plane is not essential, we have made it for convenience. They could move in any space R”, with m 3 2. 334 0022-0396/86 $3.00