Local relative integral invariants determined by the phase portrait of a vector field on the plane

We fix some invariant measure for a given vector field on the plane. The pair (the phase portrait of the vector field, the invariant measure of the vector field) determines a Lie subalgebra in the algebra of all smooth vector fields on the plane, namely the stationary subalgebra of the pair. An element of the subalgebra has a relative integral invariant, namely: the integral of the above measure along sets bounded by phase curves of the initial vector field and the element of subalgebra. The main result of the paper is the following: