  We consider inclusion of interactions between 3d Einstein gravity and the
third order extensions of Chern-Simons. Once the gravity is minimally included
into the third order vector field equations, the theory is shown to admit a
two-parameter series of symmetric tensors with on-shell vanishing covariant
divergence. The canonical energy-momentum is included into the series. For a
certain range of the model parameters, the series include the tensors that meet
the weak energy condition, while the canonical energy is unbounded in all the
instances. Because of the on-shell vanishing covariant divergence, any of these
tensors can be considered as an appropriate candidate for the right hand side
of Einstein's equations. If the source differs from the canonical energy
momentum, the coupling is non-Lagrangian while the interaction remains
consistent with any of the tensors. We reformulate these not necessarily
Lagrangian third order equations in the first order formalism which is
covariant in the sense of 1+2 decomposition. After that, we find the Poisson
bracket such that the first order equations are Hamiltonian in all the
instances, be the original third order equations Lagrangian or not. The
brackets differ from canonical ones in the matter sector, while the gravity
admits the usual PB's in terms of ADM variables. The Hamiltonian constraints
generate lapse, shift and gauge transformations of the vector field with
respect to these Poisson brackets. The Hamiltonian constraint, being the lapse
generator, is interpreted as strongly conserved energy. The matter contribution
to the Hamiltonian constraint corresponds to 00-component of the tensor
included as a source in the right hand side of Einstein equations. Once the
00-component of the tensor is bounded, the theory meets the usual sufficient
condition of classical stability, while the original field equations are of
third order.

High Energy Physics - Theory;General Relativity and Quantum Cosmology