  We present a general algorithm for constructing the holographic dictionary
for Lifshitz and hyperscaling violating Lifshitz backgrounds for any value of
the dynamical exponent $z$ and any value of the hyperscaling violation
parameter $\theta$ compatible with the null energy condition. The objective of
the algorithm is the construction of the general asymptotic solution of the
radial Hamilton-Jacobi equation subject to the desired boundary conditions,
from which the full dictionary can be subsequently derived. Contrary to the
relativistic case, we find that a fully covariant construction of the
asymptotic solution for running non-relativistic theories necessitates an
expansion in the eigenfunctions of two commuting operators instead of one. This
provides a covariant but non-relativistic grading of the expansion, according
to the number of time derivatives.
