  More that half a century ago R. Thom asserted in an unpublished manuscript
that, generically, vector fields on compact connected smooth manifolds without
boundary can admit only trivial continuous first integrals. Though somehow
unprecise for what concerns the interpretation of the word \textquotedblleft
generically\textquotedblright, this statement is ostensibly true and is
nowadays commonly accepted. On the other hand, the (few) known formal proofs of
Thom's conjecture are all relying to the classical Sard theorem and are thus
requiring the technical assumption that first integrals should be of class
$C^{k}$ with $k\geq d,$ where $d$ is the dimension of the manifold. In this
work, using a recent nonsmooth extension of Sard theorem we establish the
validity of Thom's conjecture for locally Lipschitz first integrals,
interpreting genericity in the $C^{1}$ sense.
