  In this paper we study the interplay between complex coordinates on the
Calabi-Yau metric cone and the special Killing forms on the toric
Sasaki-Einstein manifold. In the general case we give a procedure to locally
construct the special Killing forms. In the final part we exemplify the general
scheme in the case of the $5-$dimensional $Y^{p,q}$ spaces, identifying the
additional special Killing 2-forms which were previously obtained by the second
author of the present paper, but with a different method, in [Mod. Phys. Lett.
A 27 (2012) 1250217].
