|In this paper, we consider optimal resolution of air traffic conflicts. Aircraft are assumed to cruise within a given altitude layer, and are modeled as a kinematic system with velocity constraints and curvature bounds. Aircraft can not get closer to each other than a predefined safety distance. For such system of multiple aircraft, we consider the problem of planning optimal paths among given waypoints. Necessary conditions for optimality of solutions are derived, and used to devise a parameterization of possible trajectories that turns into efficient numerical solutions to the problem. Simulation results for a simplified aircraft conflict scenario are provided.