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We investigate the deformations of the Sasaki&#8722;Einstein structures of the five-dimensional spaces <inline-formula> <math display="inline"> <semantics> <msup> <mi mathvariant="normal">T</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msup> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msup> <mi mathvariant="normal">Y</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> </msup> </semantics> </math> </inline-formula> by exploiting the transverse structure of the Sasaki manifolds. We consider local deformations of the Sasaki structures preserving the Reeb vector fields but modify the contact forms. In this class of deformations, we analyze the transverse K&#228;hler&#8722;Ricci flow equations. We produce some particular explicit solutions representing families of new Sasakian structures.