  We introduce a direct generalization of the Weinstein conjecture to closed,
Lichnerowicz exact, locally conformally symplectic manifolds, (for short $\lcs$
manifolds). This conjectures existence of certain 2-curves in the manifold,
which we call Reeb 2-curves. The conjecture readily holds for all closed exact
lcs surfaces. In higher dimensions, we give partial verifications of this
conjecture, based on certain extended ($\mathbb{Q} ^{} \sqcup \{\pm \infty\}$
valued) Gromov-Witten, elliptic curve counts in $\lcs$ manifolds. As a basic
application we get some novel results in classical Reeb dynamics. The most
basic such result gives sufficient conditions for a strict contactomorphism to
fix the image of some closed Reeb orbit on a closed contact manifold. Along the
way we give a Gromov-Witten theoretic construction of the classical dynamical
Fuller index (for Reeb vector field), which among other things explains its
rationality.
