Applications of the group analysis of differential equations to some systems of noncommuting $C^1$-smooth vector fields

Given a canonical basis of $C^1$-smooth vector fields $\{\widetilde X_i\}$ satisfying certain restrictions on commutators, we prove an existence theorem for their local nilpotent homogeneous approximation at the origin using the methods of the group analysis of differential equations. We study the properties of the quasimetrics induced by some systems of vector fields related to $\{\widetilde X_i\}$.