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We contrast the dynamics of the Horava theory with anisotropic Weyl symmetrywith the case when this symmetry is explicitly broken, which is called thekinetic-conformal Horava theory. To do so we perform the canonical formulationof the anisotropic conformal theory at the classical level with a generalconformal potential. Both theories have the generator of the anisotropic Weyltransformations as a constraint but it changes from first to second-class whenthe Weyl symmetry is broken. The FDiff-covariant vector a_i = \partial_i \ln Nplays the role of gauge connection for the anisotropic Weyl transformations. ALagrange multiplier plays also the role of gauge connection, being the timecomponent. The anisotropic conformal theory propagates the same number ofdegrees of freedom of the kinetic-conformal theory, which in turn are the sameof General Relativity. This is due to exchange of a second-class constraint inthe kinetic-conformal case by a gauge symmetry in the anisotropic conformalcase. An exception occurs if the conformal potential does not depend on thelapse function N, as is the case of the so called Cotton^2 potential, in whichcase one of the physical modes becomes odd. We develop in detail two explicitanisotropic conformal models. One of them depends on N whereas the other one isthe Cotton^2 model. We also study conformally flat solutions in the anisotropicconformal and the kinetic-conformal theories, defining as conformally flat thespatial metric, but leaving for N a form different to the one dictated by theanisotropic Weyl transformations. We find that in both theories theseconfigurations have vanishing canonical momentum and they are critical pointsof the potential. In the kinetic-conformal theory we find explicitly an exact,nontrivial, conformally flat solution.

Dynamics of the anisotropic conformal Horava theory versus its kinetic-conformal formulation