Isotopy classification of Morse polynomials of degree four on ${\mathbb R}^2$

We introduce a system of invariants of isotopy classes of Morse polynomials ${\mathbb R}^2 \to {\mathbb R}^1$, prove its completeness for polynomials of degrees $\leq 4$, calculate all $71$ possible values of these invariants for the case of degree $4$, and realize them by concrete Morse polynomials. Also we calculate the number of classes (up to isotopy and reflections in ${\mathbb R}^2$) of strictly Morse polynomials of degree four with the maximal possible number of real critical points.