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We present a hypothetically complete list of 71 rigid isotopy classes ofMorse polynomials ${\mathbb R}^2 \to {\mathbb R}^1$ of degree 4 withnon-degenerate principal homogeneous part. A system of topological invariantsseparating all these classes is described. Also, we demonstrate many examplesof non-typical adjacencies of multisingularity classes to smooth functionsingularities of class $X_9$, which prevent the equisingularity of bifurcationsets of functions along the $\mu=\mbox{const} $ stratum of type $X_9$

To the rigid isotopy classification of Morse polynomials of degree 4 in ${\mathbb R}^2$