On mutual arrangements of two $M$-curves of degree $4$

We consider the problem of topological classification of mutual arrangements in the real projective plane of two $M$-curves of degree $4$. We study arrangements under the maximality condition (the oval of one of these curves has $16$ pairwise distinct common points with the oval of the other curve) and some combinatorial condition to select a special type of such arrangements. We list pairwise different topological models of arrangements of this type that satisfy the topological consequences of Bezout's theorem. There are more than 2000 such models. Examples of curves of degree $8$ realizing some of these models are given; we prove that 1728 models cannot be realized by curves of degree $8$. Proofs of the nonrealizability are performed out by Orevkov's method based on the theory of braids and links.