We examine the differential system of the unperturbed motion [1,2] with nonlinearities of degree four s3(1, 4), written in the tensorial form [3,4] x˙ j = aj αxα + aj αβγδxαxβxγxδ (j, α, β,γ, δ = 1, 2, 3) (1) where aj αβγδ is a symmetric tensor in lower indices in which the total convolution is done. The characteristic equation of this system is ϱ3 + L1ϱ2 + L2ϱ + L3 = 0, where the coefficients of this equation are expressed by center-affine invariants θ1 = aαα , θ2 = aαβ aβ α, θ3 = aαγ aβ αaγ β, from [5], and have the form L1 = −θ1, L2 = 1 2 (θ2 − θ2 1), L3 = − 1 6 (θ3 1 − 3θ1θ2 + 2θ3). (2) Using the Lyapunov’s theorems on stability of unperturbed and perturbed motion in the first approximation [2], and the Hurwitz’s theorem [2], we obtain the following theorems: Theorem 1. Assume that the center-affine invariants (2), of the system (1), satisfies the inequalities L1 > 0, L2 > 0, L3> 0, L1 L2 −L3 > 0, then the unperturbed motion x1 = x2 = x3 = 0, of the system (1), is asymptotically stable. Theorem 2. If at least one of the center-affine invariant expressions (2), of the system (1), at least one of them with the sign less than zero will be found, then the unperturbed motion x1 = x2 = x3 = 0, of the system (1), is unstable. Theorem 3 [1,2]. If for the equations of the perturbed motion can be found a function V (x) = V (x1,x2,x3), of determined sign, its derivative ˙V , would be of constant sign opposite to the sign of the function V , or identical zero, then the unperturbed motion is unstable. By a center-affne transformation, the system (1) can be brought to the critical Lyapunov form [1] and in the center-affine conditions σ1 = aαμ aβ δ aγ αxδxμxνεβγν ̸≡ 0, η1 = aαβ γδμxβxγxδxμxνyθεανθ ≡ 0, L1,L2 > 0, the system (1) becomes a critical of Lyapunov-Darboux type, of the form    x˙ = −λy + 4xR(x, y, z) , y˙ = λx + 4yR(x, y, z) , z˙ = y − L1z + 4zR(x, y, z) , (3) where x = x1, y = x2, z = x3, R(x, y, z) = a1x3 + a2y3 + a3z3 + 3a4x2y+ +3a5x2z + 3a6xy2 + 3a7xz2 + 3a8xyz + 3a9y2z + 3a10yz2, and ai (i = 1, 10) are coefficients that takes values from the fields of real numbers R. Using Lie algebra, of the system (3), we obtain the analytic first integral of the form F(x, y, z) ≡ h31 (J + h2)2 = C (4) governed by the condition J(J + h2) ̸= 0, (5) where h1 = x2 + y2, J = −L1λ2(4L21 + λ2)(L21 + 4λ2), h2 = λ[4(8a3L21 + 24a10L31 + 12a5L41 + 24a9L41 + 8a2L51 + +12a4L51 − 24a7L21 λ − 12a8L31 λ + 22a3λ2 + 66a10L1λ2+ +75a5L21 λ2 + 78a9L21 λ2 + 34a2L31 λ2 + 51a4L31 λ2− −36a7λ3 − 3a8L1λ3 + 18a5λ4 + 18a9λ4 + 8a2L1λ4+ +12a4L1λ4)x3 − 4L1(12a7L21 + 12a8L31 + 8a1L41 + 12a6L41 + +10a3λ + 30a10L1λ − 24a5L21 λ + 24a9L21 λ − 12a7λ2 + 3a8L1λ2+ +34a1L21 λ2 + 51a6L21 λ2 − 6a5λ3 + 6a9λ3 + 8a1λ4 + 12a6λ4)y3+ +4a3λ(4L21+ λ2)(L21 + 4λ2)z3 − 12a1L1(4L21 + λ2)(L21 + +4λ2)x2y + 12λ(12a5L41 + 12a7L21 λ + 12a8L31 λ + 10a3λ2+ +30a10L1λ2 + 27a5L21 λ2 + 24a9L21 λ2 − 12a7λ3 + 3a8L1λ3++6a5λ4 + 6a9λ4)x2z + 12(4a3L21 + 12a10L31 + 12a9L41 + 4a2L51 − −18a7L21 λ − 12a8L31 λ + 6a3λ2 + +18a10L1λ2 + 24a5L21 λ2+ +27a9L21 λ2 + 17a2L31 λ2 − 12a7λ3 − 3a8L1λ3 + 6a5λ4 + 6a9λ4+ +4a2L1λ4)xy2 + 12λ(6a7L21 + a3λ + 3a10L1λ)(L21 + 4λ2)xz2+ +12L1λ(12a7L21 + 12a8L31 + 10a3λ + 30a10L1λ − 24a5L21 λ+ +24a9L21 λ − 12a7λ2 + 3a8L1λ2 − 6a5λ3 + 6a9λ3)xyz+ +12λ(4a3L21 + 12a10L31 + 12a9L41 − 18a7L21 λ − 12a8L31 λ + 6a3λ2+ +18a10L1λ2 + 24a5L21 λ2 + 27a9L21 λ2 − 12a7λ3 − 3a8L1λ3 + 6a5λ4+ +6a9λ4)y2z + 12L1λ(2a3 + 6a10L1 − 3a7λ)(L21 + 4λ2)yz2]. Analyzing the first integral (4), we notice that if the inequality (5) holds, when the function F(x, y, z), from (4), forms the Lyapunov function. Then according to the theorem 3, we have Theorem 4. If for the system of Lyapunov-Darboux type (3) the inequality (5) holds, then the unperturbed motion x = y = z = 0, governed by this system is stable.