  Recently we studied `vanishing' horizon limits of `boosted' black D3-brane
geometry \cite{hsnr}. The type IIB solutions obtained by taking these special
double limits were found to describe nonrelativistic Lifshitz spacetimes at
zero temperature. In the present work we study these limits for TsT black-hole
solutions which include $B$-field. The new Galilean solutions describe a
holographic RG flow from Schr\"odinger ($a=2$) spacetime in UV to a Lifshitz
universe ($a=3$) in the IR.
