Three-webs $W(1,n,1)$ and associated systems of ordinary differential equations

We consider a three-web formed by two $n$-parameter families of curves and an one-parameter family of hypersurfaces on a smooth manifold. For such webs we define a family of adapted frames, formulate a system of structural equations, and study differential-geometric objects that arise in differential neighborhoods up to the third order. We prove that each system of ordinary differential equations (SODE) uniquely defines some three-web. This allows us to describe properties of SODE in terms of the corresponding three-web. In particular, we characterize autonomous SODE.