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In this paper we investigate the possibility to obtain locally newSasaki-Einstein metrics on the space $T^{1,1}$ considering a deformation of thestandard metric tensor field. We show that from the geometric point of viewthis deformation leaves transverse and the leafwise metric intact, but changesthe orthogonal complement of the Reeb vector field using a particular basicfunction. In particular, the family of metric obtained using this method can beregarded as solutions of the equation associated to the Sasaki-Ricci flow onthe underlying manifold.

Transverse K\"{a}hler-Ricci flow and deformations of the metric on the Sasaki space $T^{1,1}$