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In the present paper we show that the complete list of special Killing formson the 5-dimensional Sasaki-Einstein spaces $Y^{p,q}$ can be extracted usingthe symplectic potential and the classical Delzant construction. The resultsachieved here agree with previous ones obtained by direct computation, provingthe reliability of the method which stands in fact as a general algorithm fortoric Sasaki-Einstein manifolds. Finally, we discuss the integrability ofgeodesic motion in $Y^{p,q}$ spaces.

Toric data, Killing forms and complete integrability of geodesics in Sasaki-Einstein spaces $Y^{p,q}$