  We construct explicitly the constants of motion for geodesics in the
$5$-dimensional Sasaki-Einstein spaces $Y^{p,q}$. To carry out this task we use
the knowledge of the complete set of Killing vectors and Killing-Yano tensors
on these spaces. In spite of the fact that we generate a multitude of constants
of motion, only five of them are functionally independent implying the complete
integrability of geodesic flow on $Y^{p,q}$ spaces. In the particular case of
the homogeneous Sasaki-Einstein manifold $T^{1,1}$ the integrals of motion have
simpler forms and the relations between them are described in detail.
