Strict contactomorphisms are scarce

The notion of non-projectible contact forms on a given compact manifold $M$ is introduced by the first-named author in [Ohb], the set of which he also shows is a residual subset of the set of (coorientable) contact forms, both in the case with a fixed contact structure and in the case without it. In this paper, we prove that for any non-projectible contact form $\lambda$ the set, denoted by $\text{\rm Cont}^{\text{\rm st}}(M,\lambda)$, consisting of strict contactomorphisms of $\lambda$ is a a countable disjoint union of real lines $\mathbb R$, one for each connected component.