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In this paper, we consider the multi-species nonlinear Schr\"odinger systemsin $\bbr^N$:  \begin{equation*}  \left\{\aligned&-\Deltau_j+V_j(x)u_j=\mu_ju_j^3+\sum_{i=1;i\not=j}^d\beta_{i,j} u_i^2u_j\quad\text{in}\bbr^N,  &u_j(x)>0\quad\text{in } {\mathbb R}^N,  &u_j(x)\to0\quad\text{as }|x|\to+\infty,\quadj=1,2,\cdots,d,\endaligned\right.  \end{equation*}  where $N=2,3$, $\mu_j>0$ are constants, $\beta_{i,j}=\beta_{j,i}\not=0$ arecoupling parameters, $d\geq2$ and $V_j(x)$ are potentials. By Ljapunov-Schmidtreduction arguments, we construct infinitely many nonradial positive solutionsof the above system under some mild assumptions on potentials $V_j(x)$ andcoupling parameters $\{\beta_{i,j}\}$, {\it without any symmetric assumptionson the limit case of the above system}. Our result, giving a positive answer tothe conjecture in Pistoia and Vaira \cite{PV22} and extending the results in\cite{PW13,PV22}, reveals {\it new phenomenon} in the case of $N=2$ and $d=2$and is {\it almost optimal} for the coupling parameters $\{\beta_{i,j}\}$.

Infinitely many nonradial positive solutions for multi-species nonlinear Schr\"odinger systems in ${\mathbb R}^N$