Patchworking singularities $A_\mu$ and $D_\mu$ and meanders of their smoothing

Let an algebraic curve $f$ have a singular point of type $A_{\mu}$ or $D_{\mu}$. Let $\tilde{f}$ be the curve obtained as a result of smoothing the singular point of the curve $f$. In this paper we study the local maximal meanders appearing under $M$-smoothing in a neighborhood of the singular point. A local maximal meander means that the number of real points of the intersection of the curve $\tilde{f}$ with a coordinate axis in the neighborhood is maximal and the points belong to one of the components of $\tilde{f}$; and an $M$-smoothing means that the number of components of the curve $\tilde{f}$, which appear in the neighborhood under the smoothing, is also maximal.