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We demonstrate how the novel approach to the local geometry of structures ofnonholonomic nature, originated by Andrei Agrachev, works in the following twosituations: rank 2 distributions of maximal class in R^n with non-zerogeneralized Wilczynski invariants and rank 2 distributions of maximal class inR^n with additional structures such as affine control system with one inputspanning these distributions, sub-(pseudo)Riemannian structures etc. The commonfeature of these two situations is that each abnormal extremal (of theunderlying rank 2 distribution) possesses a distinguished parametrization. Thisfact allows one to construct the canonical frame on a (2n-3)-dimensional bundlein both situations for arbitrary n greater than 4. The moduli spaces of themost symmetric models for both situations are described as well. The relationof our results to the divergence equivalence of Lagrangians of higher order isgiven

Geometry of rank 2 distributions with nonzero Wilczynski invariants and affine control systems with one input