  In this article we study maps with nilpotent Jacobian in $\mathbb{R}^n$
distinguishing the cases when the rows of $JH$ are linearly dependent over
$\mathbb{R}$ and when they are linearly independent over $\mathbb{R}.$ In the
linearly dependent case, we show an application of such maps on dynamical
systems, in particular, we construct a large family of almost Hurwitz vector
fields for which the origin is an almost global attractor. In the linearly
independent case, we show explicitly the inverse maps of the counterexamples to
Generalized Dependence Problem and proving that this inverse maps also have
nilpotent Jacobian with rows linearly independent over $\mathbb{R}.$
