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In this paper we focus on existence and symmetry properties of solutions tothe cubic Schr\"odinger system \[ -\Delta u_i +\lambda_i u_i = \sum_{j=1}^d\beta_{ij} u_j^2 u_i \quad \text{in $\Omega \subset \mathbb{R}^N$},\qquadi=1,\dots d \] where $d\geq 2$, $\lambda_i,\beta_{ii}>0$,$\beta_{ij}=\beta_{ji}\in \mathbb{R}$ for $j\neq i$, $N=2,3$. The underlyingdomain $\Omega$ is either bounded or the whole space, and $u_i\inH^1_0(\Omega)$ or $u_i\in H^1_{rad}(\mathbb{R}^N)$ respectively. We establishnew existence and symmetry results for least energy positive solutions in thecase of mixed cooperation and competition coefficients, as well as in thepurely cooperative case.

New existence and symmetry results for least energy positive solutions of Schr\"odinger systems with mixed competition and cooperation terms