We obtain a solution of the Chazy system that consists of nine nonlinear algebraic equations. This system gives a necessary condition for the class of nonlinear third-order differential equations with six singularities to be of P -type.

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Consider the autonomous differential equation y 3 y IV = a 1 y 2 y y + a 2 y 2 y 2 + a 3 yy 2 y + a 4 y 4 + a 5 y 4 y + a 6 y 3 y y + a 7 y 2 y 3 + a 8 y 5 y + a 9 y 4 y 2 + a 10 y 6 y + a 11 y 8 .A method for finding first integrals based on the existence of roots of specific form for the resonance equations for polynomial differential equations was suggested in [1,2]. Similar considerations are used in the present paper for finding sufficient conditions for the existence of a first integral of Eq. (1). Under the assumption that Eq. (1) has no solutions with logarithmic singularities, we seek a solution in the form of the seriesBy substituting the expression (2) into Eq. (1), for h 0 and the resonance r, we obtain the equationswhereand the h 0i , i = 1, 2, 3, 4, are distinct roots of Eq. (3).The following sufficient conditions for the existence of first integrals were obtained in [3] for the case in which a i = 0, i = 5, . . . , 11, and |a 1 | + |a 2 | + |a 3 | + |a 4 | = 0 :(1 0 ) a 1 = a − 4γ − 2, a 2 = 2a + (2γ + 1), a 3 = (4γ + 1)a + 4b, a 4 = 4γb;(2 0 ) a 1 = a − 3γ − 2, a 2 = a, a 3 = (3γ + 1)a + 3b, a 4 = 3γb;here a, b, γ, and δ are parameters. We assume that conditions (6) are valid for a i , i = 1, 2, 3, 4, and at least one of the coefficients a 5 , . . . , a 11 is nonzero. Consider the following cases. 1. Let a 11 = 0, (7) and let r = 4ν be a root of Eq. (4) for every h 0i , i = 1, 2, 3, 4. By setting γ = ν − 2, we rewrite condition 1 0 in (6) in the form a 1 = 6 + a − 4ν, a 2 = 2a + (2ν − 3), a 3 = (4ν − 7)a + 4b, a 4 = (4ν − 8)b.

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The object of this research is fourth-order differential equations. The aim of the research is to study the analytical properties of the solutions of these differential equations. The general form of the considered equations is indicated, and also the choice of the research object is justified. Herein we studied fourth-order differential equations for which sets of resonances with all positive nontrivial resonances are absent. Besides, three of these equations satisfy the conditions of absence in the solutions of moving multivalued singular points. The solutions of the next three equations have movable special points of multivalued character. Moreover, we also investigated the analytical properties of one more fourth-order differential equation of another general form for which it is also possible to construct a two-parameter rational solution as there is a nontrivial negative resonance in the related set of resonances. The first integrals of the equations under study are found and their rational solutions are constructed from negative non-trivial resonances. The resonance method was used in this study. The obtained results can be used in the analytical theory of differential equations.