On the existence and prolongation of infinitesimal isometries on special sub-Riemannian manifolds

In the present paper we deal with (local) infinitesimal isometries of special sub-Riemannian manifolds (a contact and oriented sub-Riemannian manifold is called special if the Reeb vector field is an infinitesimal isometry). The objective of the paper is to find some conditions on such manifolds which allow one to construct, locally around a given point, infinitesimal isometries and then, if possible, to prolong them onto bigger domains. The mentioned conditions are related to the so-called \(\mathfrak {i}^*\)-regular and \(\mathfrak {i}\)-regular points, the notions introduced by Nomizu (Ann Math 2:105–120, 1960) in the Riemannian setting and slightly modified by the author.