We consider the following nonlinear Schrodinger system in R-3 {-Delta u+P(vertical bar x vertical bar)u=mu u(3)+beta upsilon(2)u, x epsilon R-3, {-Delta upsilon+Q(vertical bar x vertical bar)upsilon=nu upsilon(3)+beta u(2)upsilon, x epsilon R-3, where P(r) and Q(r) are positive radial potentials, mu > 0, nu > 0 and beta epsilon R is a coupling constant. This type of system arises, in particular, in models in Bose-Einstein condensates theory. We examine the effect of nonlinear coupling on the solution structure. In the repulsive case, we construct an unbounded sequence of non-radial positive ve...