  Let $k$ be a differential field of characteristic zero and $E$ be a
liouvillian extension of $k$. For any differential subfield $K$ intermediate to
$E$ and $k$, we prove that there is an element in the set $K-k$ satisfying a
linear homogeneous differential equation over $k$. We apply our results to
study liouvillian solutions of first order non linear differential equations
and provide generalisations and new proofs for several results of M. Singer and
M. Rosenlicht on this topic.
