On a class of non linear differential operators of first order with singular point

We consider the problem of the existence and uniqueness of solutions for partial differential operator of the form $Lu=D_Xu-B(x,u)$ where $X$ is a vector field. The solvability of $L$ may be of some interest since by the Nash–Moser inverse function theorem the equivalence problem in differential geometry can be solved via Lie derivative operator and the later is locally a particular case of $L$. An application to the equivalence of dynamic systems is given.