Three-webs defined by symmetrical functions

We consider local differential-geometrical properties of curvilinear $k$-webs defined by symmetric functions (the webs $SW(k)$). The algebraic rectilinear $k$-webs defined by algebraic curves of genus $0$ are the symmetric $k$-webs. We prove that $3$ three-parameter families of $T$-configurations are closed on every symmetric $k$-web. We find the equations of a rectilinear $SW(k)$-web in adapted coordinates. It is proved that the curvature of a $SW(k)$-web is a skew-symmetric function with respect to adapted coordinates. In conclusion, we formulate some unsolved problems.