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A Carter like constant for the geodesic motion in the $Y(p,q)$Einstein-Sasaki geometries is presented. This constant is functionallyindependent with respect to the five known constants for the geometry. Sincethe geometry is five dimensional and the number of independent constants ofmotion is at least six, the geodesic equations are superintegrable. We pointout that this result applies to the configuration of massless geodesic in$AdS_5\times Y(p,q)$ studied by Benvenuti and Kruczenski, which are matched tolong BPS operators in the dual N=1 supersymmetric gauge theory.

Massless geodesics in $AdS_5\times Y(p,q)$ as a superintegrable system