We consider the problem of the existence and determining stationary Nash equilibria for average single-controller stochastic games. Single-controller stochastic games with average payoffs represent a class of average stochastic games in which the transition probabilities are controlled by one player only. The problem of determining stationary Nash equilibria in such games has been studied in [2–5]. In [2, 3] has been proposed a linear programming algorithm for computing stationary equilibria in the case of two-player zero-sum games. The problem of the existence and determining stationary equilibria for a more general case of single-controlled average stochastic games has been considered in [5, 6] . We propose an approach for determining stationary Nash equilibria for single-controller stochastic games with average payoffs in general case. We show that all stationary equilibria for a single controller stochastic game can be obtained from an auxiliary noncooperative static game in normal form where the payoffs are quasi-monotonic (quasi-convex and quasi-concave) with respect to the corresponding strategies of the players and graph-continuous in the sense of Dasgupta and Maskin [1, 4]. Based on this we present a proof of the existence of stationary equilibria in a single-controller average stochastic game and propose an approach for determining the optimal stationary strategies of the players.