  We study invariants defined by count of charged, elliptic $J$-holomorphic
curves in locally conformally symplectic manifolds. We use this to define
$\mathbb{Q} $-valued deformation invariants of certain complete Riemann-Finlser
manifolds and their isometries and this is used to find some new phenomena in
Riemann-Finlser geometry. In contact geometry this Gromov-Witten theory is used
to study fixed Reeb strings of strict contactomorphisms. Along the way, we
state an analogue of the Weinstein conjecture in lcs geometry, directly
extending the Weinstein conjecture, and discuss various partial verifications.
A counterexample for a stronger, also natural form of this conjecture is given.
