  In this article, we study the effects of the propagation of a non-degenerate
L\'evy noise through a chain of deterministic differential equations whose
coefficients are H\"older continuous and satisfy a weak H\"ormander-like
condition. In particular, we assume some non-degeneracy with respect to the
components which transmit the noise. Moreover, we characterize, for some
specific dynamics, through suitable counterexamples , the almost sharp
regularity exponents that ensure the weak well-posedness for the associated
SDE. As a by-product of our approach, we also derive some Krylov-type estimates
for the density of the weak solutions of the considered SDE.
