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.
