  We present some examples of locally conformal symplectic structures of the
first kind on compact nilmanifolds which do not admit Vaisman metrics. One of
these examples does not admit locally conformal K\"ahler metrics and all the
structures come from left-invariant locally conformal symplectic structures on
the corresponding nilpotent Lie groups. Under certain topological restrictions
related with the compactness of the canonical foliation, we prove a structure
theorem for locally conformal symplectic manifolds of the first kind. In the
non compact case, we show that they are the product of a real line with a
compact contact manifold and, in the compact case, we obtain that they are
mapping tori of compact contact manifolds by strict contactomorphisms.
Motivated by the aforementioned examples, we also study left-invariant locally
conformal symplectic structures on Lie groups. In particular, we obtain a
complete description of these structures (with non-zero Lee $1$-form) on
connected simply connected nilpotent Lie groups in terms of locally conformal
symplectic extensions and symplectic double extensions of symplectic nilpotent
Lie groups. In order to obtain this description, we study locally conformal
symplectic structures of the first kind on Lie algebras.
