Structure of the local controllability set for a family of $2$-systems on a plane near the zero indicatrix point

In this work it is shown that if certain natural conditions of community of position hold, then critical points of the local controllability set for a $2$-parameter family of $2$-systems on a plane near the point with zero indicatrix of the velocity are well defined by first derivatives of fields of the family at the point. A method to construct this set near such a point is given.