  In this paper, we consider the following nonlinear Schr\"odinger system:
  -$\Delta$ u+P(x)u=$\mu_1$ $u^3$+$\beta$ u$v^2$, x $\in$ $R^3$,\\ -$\Delta$
v+Q(x)v=$\mu_2$ $v^3$+$\beta$ $u^2$v, x $\in$ $R^3$,
  where $P(x),Q(x)$ are positive radial potentials,~$\mu_1,\,\mu_2>0$, $\beta$
$\in$ $R$ is a coupling constant. We constructed a new type of solutions which
are different from the ones obtained in \cite{PW}. This new family of solutions
to system have a more complex concentration structure and are centered at the
points lying on the top and the bottom circles of a cylinder with height $h$.
Moreover, we examine the effect of nonlinear coupling on the solution
structure. In the repulsive case, we construct an unbounded sequence of
non-radial positive vector solutions of segregated type. In the attractive
case, we construct an unbounded sequence of non-radial positive vector
solutions of synchronized type. Moreover, we prove that there exist infinitely
many sign-changing solutions whose energy can be arbitrarily large.
