For a generic non-projectible contact form on a compact manifold, every strict contactomorphism is a Reeb flow, so the strict contactomorphism group is a countable union of real lines.
Ebin and Jerrold Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann
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Strict contactomorphisms are scarce
For a generic non-projectible contact form on a compact manifold, every strict contactomorphism is a Reeb flow, so the strict contactomorphism group is a countable union of real lines.