  We construct solutions describing flows between AdS and Lifshitz spacetimes
in IIB supergravity. We find that flows from AdS$_5$ can approach either
AdS$_3$ or Lifshitz$_3$ in the IR depending on the values of the deformation
from AdS$_5$. Surprisingly, the choice between AdS and Lifshitz IR depends only
on the value of the deformation, not on its character; the breaking of the
Lorentz symmetry in the flows with Lifshitz IR is spontaneous. We find that the
values of the deformation which lead to flows to Lifshitz make the UV field
theory dual to the AdS$_5$ geometry unstable, so that these flows do not offer
an approach to defining the field theory dual to the Lifshitz spacetime.
