  Dynamical realizations of the Lifshitz group are studied within the
group-theoretic framework. A generalization of the 1d conformal mechanics is
constructed, which involves an arbitrary dynamical exponent z. A similar
generalization of the Ermakov-Milne-Pinney equation is proposed. Invariant
derivative and field combinations are introduced, which enable one to construct
a plethora of dynamical systems enjoying the Lifshitz symmetry. A metric of the
Lorentzian signature in (d+2)-dimensional spacetime and the energy-momentum
tensor are constructed, which lead to the generalized Ermakov-Milne-Pinney
equation upon imposing the Einstein equations. The method of nonlinear
realizations is used for building Lorentzian metrics with the Lifshitz isometry
group. In particular, a (2d+2)-dimensional metric is constructed, which enjoys
an extra invariance under the Galilei boosts.
