  We consider conformal actions of solvable Lie groups on closed Lorentzian
manifolds. With anterior results in which we addressed similar questions for
semi-simple Lie group actions, this work contributes to the understanding of
the identity component $G$ of the conformal group of closed Lorentzian
manifolds. In the first part of the article, we prove that $G$ is inessential
if and only if its nilradical is inessential. In the second, we assume the
nilradical essential and establish conformal flatness of the metric on an open
subset, under certain algebraic hypothesis on the solvable radical. This is
related to the Lorentzian Lichnerowicz conjecture. Finally, we consider the
remaining situations where our methods do not apply to prove conformal
flatness, and conclude that for an essential closed Lorentzian $n$-manifold, $n
\geq 3$, the radical of its conformal group admits a local embedding into
$O(2,n)$.
