  Under the uniform H\"{o}rmander's hypothesis we study smoothness and
exponential bounds of the density of the law of the solution of a stochastic
differential equation (SDE) with locally Lipschitz drift that satisfy a
monotonicity condition. To avoid non-integrability problems we use results
about Malliavin differentiability based on the concepts of Ray Absolute
Continuity and Stochastic Gate\^aux differentiability.
