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We have studied the existence of self-dual solitonic solutions in ageneralization of the Abelian Chern-Simons-Higgs model. Such a generalizationintroduces two different nonnegative functions, $\omega_1(|\phi|)$ and$\omega(|\phi|)$, which split the kinetic term of the Higgs field -$|D_\mu\phi|^2 \rightarrow\omega_1 (|\phi|)|D_0\phi|^2-\omega(|\phi|)|D_k\phi|^2$ - breaking explicitly the Lorentz covariance. We have shown that aclean implementation of the Bogomolnyi procedure only can be implementedwhether $\omega(|\phi|) \propto \beta |\phi|^{2\beta-2}$ with $\beta\geq 1$.The self-dual or Bogomolnyi equations produce an infinity number of solitonsolutions by choosing conveniently the generalizing function $\omega_1(|\phi|)$which must be able to provide a finite magnetic field. Also, we have shown thatby properly choosing the generalizing functions it is possible to reproduce theBogomolnyi equations of the Abelian Maxwell-Higgs and Chern-Simons-Higgsmodels. Finally, some new self-dual $|\phi|^6$-vortex solutions have beenanalyzed both from theoretical and numerical point of view.

Self-dual configurations in a generalized Abelian Chern-Simons-Higgs model with explicit breaking of the Lorentz covariance