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We prove, using variational methods, the existence in dimension two ofpositive vector ground states solutions for the Bose-Einstein type systems\begin{equation} \begin{cases} -\Delta u+\lambda_1u=\mu_1u(e^{u^2}-1)+\betav\left(e^{uv}-1\right) \text{ in } \Omega, &\\ -\Deltav+\lambda_2v=\mu_2v(e^{v^2}-1)+\beta u\left(e^{uv}-1\right)\text{ in } \Omega,&\\ u,v\in H^1_0(\Omega) \end{cases} \end{equation} where $\Omega$ is a boundedsmooth domain, $\lambda_1,\lambda_2>-\Lambda_1$ (the first eigenvalue of$(-\Delta,H^1_0(\Omega))$, $\mu_1,\mu_2>0$ and $\beta$ is either positive(small or large) or negative (small). The nonlinear interaction between twoBose fluids is assumed to be of critical exponential type in the sense of J.Moser. For `small' solutions the system is asymptotically equivalent to thecorresponding one in higher dimensions with power-like nonlinearities.

Bose fluids and positive solutions to weakly coupled systems with critical growth in dimension two