On contact Hamiltonian functions of singular contact structures

For contact manifolds, it is well-known that the map which assigns to an infinitesimal contact transformation its contact Hamiltonian function is a linear isomorphism, and an explicit local formula for its inverse can be given. In contrast, infinitesimal contact transformations on a singular contact structure have not been well-studied yet. In this article, we show that this map is injective and present an explicit local formula for its inverse when the contact structure has singularities of the first type with structurally smooth Martinet hypersurface.