Abstract
We propose homogeneous metrics of Petrov type III that describe gyrating Schr\"odinger geometries as duals to some nonrelativistic field theories, in which the Schr\"odinger symmetry is broken further so that the phase space has a linear dependence of the momentum in a selected direction. We show that such solutions can arise in four-dimensional Einstein-Weyl supergravity as well as higher-dimensional extended gravities with quadratic curvature terms coupled to a massive vector. In Einstein-Weyl supergravity, the gyrating Schr\"odinger solutions can be supersymmetric, preserving $\frac{1}{4}$ of the supersymmetry. We obtain the exact Green function in the phase space associated with a bulk free massive scalar.
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