On the dispositions of two $M$-singular curves of degree $4$, the oval of one of which coils around the oval of the other

We consider the problem of topological classification of mutual dispositions in the real projective plane of two $M$-curves of degree $4$. We study arrangements which are satisfact to the maximality condition (the oval of one of these curves has $16$ pairwise different common points with the oval of the other of them) and some combinatorial condition to select a special type of such arrangements. Pairwise different topological models of arrangements of this type are listed, which satisfy the known facts about the topology of nonsingular curves and the topological consequences of Bezout's theorem. There are $564$ such models. We proved that $558$ models cannot be realized by curves of degree $8$. The remaining $6$ models were constructed by us. Proofs of non-realizability are carried out by Orevkov's method based on the theory of braids and links.