Curves in a Foliated Plane

The paper is devoted to the classification of nonsingular and singular plane curve germs with respect to the group of local diffeomorphisms preserving the foliation of the plane by the phase curves of a fixed vector field, either nonsingular or singular. We define the multiplicity of a pair consisting of a plane curve and a vector field and prove an analog of the Tougeron theorem on finite determinacy. It leads, almost immediately, to a number of classification results; a part of them is contained in the work.