Special polynomials associated with the fourth order analogue to the Painlevé equations

Rational solutions of the fourth order analogue to the Painlevé equations are classified. Special polynomials associated with the rational solutions are introduced. The structure of the polynomials is found. Formulae for their coefficients and degrees are derived. It is shown that special solutions of the Fordy–Gibbons, the Caudrey–Dodd–Gibbon and the Kaup–Kupershmidt equations can be expressed through solutions of the equation studied.