We consider the system of perturbed Schr(o)dinger equations{- ε2△(ψ) + α(x)(ψ) = β(x)ψ + Fψ(x, (ψ), ψ)- ε2△ψ + α(x)ψ = β(x)(ψ) + F(ψ)(x,(ψ),ψ)w := ((ψ),ψ) ∈H1(RN,R2)where N ≥ 1, α and β are continuous real functions on RN, and F: RN × R2 → R is of class C1. We assume that either F(x, w) is super-quadratic and subcritical in w ∈ R2 or it is of the form ～ 1/p P(x)|w|p + 1/2*K(x)|w|2* with p ∈ (2,2*) and 2* = 2N/(N - 2),the Sobolev critical exponent, P(x) and K(x) are positive bounded functions. Under proper conditions we show that the system has at least one nontrivial solution wε provided ε≤ε; and for any m ∈ N, there are m pairs of solutions wε provided that ε≤εm and that F(x, w) is,in addition, even in w. Here ε and εm are sufficiently small positive numbers. Moreover, the energy of wε tends to 0 as ε→ 0.