Normalized solutions to critical Choquard systems with linear and nonlinear couplings

We consider the critical Choquard system with both linear and nonlinear couplings $-\Delta v_1 + \mu_1 v_1 = ( I_\omega * |v_1|^{2_\omega^*} ) |v_1|^{2_\omega^* -2} v_1 + \theta p( I_\omega * |v_2|^q)|v_1|^{p-2} v_1 + \varepsilon v_2, \quad in \,\, \mathbb{R}^N, -\Delta v_2 + \mu_2 v_2 = ( I_\omega * |v_2|^{2_\omega^*} ) |v_2|^{2_\omega^* -2} v_2 + \theta q( I_\omega * |v_1|^p)|v_2|^{q-2} v_2 + \varepsilon v_1 , \quad in \,\, \mathbb{R}^N , \int_{\mathbb{R}^N} v_1^2 = \alpha_1^2\, , \int_{\mathbb{R}^N} v_2^2 = \alpha_2^2,$ where $N=3\,\, \text{or} \,\, 4$, $\alpha_1,\alpha_2 > 0$, $\theta > 0$, $2_{\omega,*} :=\frac{N+\omega}{N} <p,q<2_\omega^*:=\frac{N+\omega}{N-2}$, $\varepsilon>0$, $0<\omega<N$, $I_\omega: \mathbb{R}^N \to \mathbb{R}$ represents the Riesz potential. For the $L^2$-subcritical case $p+q<\frac{2N+2\omega+4}{N}$, we utilize the Ekeland's variational principle to obtain the existence of a positive normalized ground state for the system as $0<\theta<\theta_0,\;0<\varepsilon<\varepsilon_*$. For the $L^2$-supercritical case $p+q>\frac{2N+2\omega+4}{N}$, we apply variational methods to establish the existence of a positive normalized ground state for the system as $\theta>\theta_*,\;0<\varepsilon<\overline{\varepsilon}$.