  We prove, using variational methods, the existence in dimension two of
positive vector ground states solutions for the Bose-Einstein type systems
\begin{equation} \begin{cases} -\Delta u+\lambda_1u=\mu_1u(e^{u^2}-1)+\beta
v\left(e^{uv}-1\right) \text{ in } \Omega, &\\ -\Delta
v+\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 bounded
smooth 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 two
Bose fluids is assumed to be of critical exponential type in the sense of J.
Moser. For `small' solutions the system is asymptotically equivalent to the
corresponding one in higher dimensions with power-like nonlinearities.
