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We give several sufficient conditions under which the first-order nonlinear Hamiltonian system x'(t)=α(t)x(t)+f(t,y(t)),  y'(t)=-g(t,x(t))-α(t)y(t) has no solution (x(t),y(t)) satisfying condition 0<∫-∞+∞[|x(t)|ν+(1+β(t))|y(t)|μ]dt<+∞‍, where μ,ν>1 and (1/μ)+(1/ν)=1, 0≤xf(t,x)≤β(t)|x|μ, xg(t,x)≤γ0(t)|x|ν, β(t),γ0(t)≥0, and α(t) are locally Lebesgue integrable real-valued functions defined on ℝ.