  Taking as a starting point a Lorentz non-invariant Abelian-Higgs model
defined in 1+3 dimensions, we carry out its dimensional reduction to D=1+2,
obtaining a new planar model composed by a Maxwell-Chern-Simons-Proca gauge
sector, a massive scalar sector, and a mixing term (involving the fixed
background (v^{\mu}) that imposes the Lorentz violation to the reduced model.
The propagators of the scalar and massive gauge field are evaluated and the
corresponding dispersion relations determined. Based on the poles of the
propagators, a causality and unitarity analysis is carried out at tree-level.
One then shows that the model is totally causal and unitary.
