We study the problem of the existence of a Green-Samoilenko function for some linear extensions of dynamical systems.Consider the system of differential equationswith real coefficients a, b, a j , b j , j = 0, n, i = 1, n. The problem is to find conditions on these coefficients under which system (1) has infinitely many different Green-Samoilenko functions. Recall (see [1]) that, for theGreen-Samoilenko function is a function of the formprovided that, for a certain continuous n×n matrix C(ϕ) ∈ C(T m ), the following estimate is true: G 0 (τ, ϕ) ≤ K exp{−γ|τ |}, where K and γ are positive constants independent of τ ∈ R and ϕ ∈ T m . Here, ϕ τ (ϕ) is the solution of the Cauchy problem dϕ dt = a(ϕ), ϕ t (ϕ) t=0 = ϕ, Ω t τ (ϕ) is the matrizant of the linear system dx dt = A(ϕ t (ϕ))x normalized at t = τ, and Ω t τ (ϕ) t=τ = I n is the identity matrix. Interesting fundamental investigations concerning the existence and properties of the function G 0 (τ, ϕ) were carried out in [2, 3]. However, even in the scalar case (n = 1, m = 1) there exist systems for which the problem of the existence of a GreenSamoilenko G 0 (τ, ϕ) function remains open. For system (1), the problem of the existence of a function G 0 (τ, ϕ) is solved for arbitrary fixed real numbers a, b, a j , b j , j = 0, n, i = 1, n, and the following cases are analyzed in detail: (i) a Green-Samoilenko function is unique, (ii) there are infinitely many Green-Samoilenko functions, and (iii) a Green-Samoilenko function does not exist. In the present paper, we continue the investigation of these problems.