On some local bifurcations of reversible piecewise smooth dynamical systems on the plane

A two-parameter family of piecewise-smooth vector fields on the plane, “sewn” from smooth vector fields defined in the upper and lower half-planes, is considered. The vector fields of the family are assumed to be reversible with respect to an inversion for which the line of discontinuity of the field $y = 0$ consists of fixed points. At zero values of the parameters, the vector fields defined in the upper and lower half-planes have a third-order tangency with the $x$-axis at the origin of coordinates $O$. Bifurcations of phase portraits in a neighborhood of point $O$ are described for parameter values close to zero.