We study the interplay between the properties of the germ of a
singular variety $N\subset \mathbb R^n$ given in the title and the
algebra of vector fields tangent to $N$. The Poincare lemma
property means that any closed differential $(p+1)$-form vanishing
at any point of $N$ is a differential of a $p$-form which also
vanishes at any point of $N$. In particular, we show that the
classical quasi-homogeneity is not a necessary condition for the
Poincare lemma property; it can be replaced by quasi-homogeneity
with respect to a smooth submanifold of $\mathbb R^n$ or a chain
of smooth submanifolds. We prove that $N$ is quasi-homogeneous if
and only if there exists a vector field $V, V(0)=0,$ which is
tangent to $N$ and has positive eigenvalues. We also generalize
this theorem to quasi-homogeneity with respect to a smooth
submanifold of $\mathbb R^n$.