Normalized solutions to critical Choquard systems with linear and nonlinear couplings

We consider the critical Choquard system with both linear and nonlinear couplings $-\mathrm{\Delta }{v}_1+{\mu }_1{v}_1=(I_{\omega }\ast \mathrm{\mid }{v}_1{\mathrm{\mid }}^{2_{\omega }^{\ast }})\mathrm{\mid }{v}_1{\mathrm{\mid }}^{2_{\omega }^{\ast }-2}{v}_1+\theta p(I_{\omega }\ast \mathrm{\mid }{v}_2{\mathrm{\mid }}^q)\mathrm{\mid }{v}_1{\mathrm{\mid }}^{p-2}{v}_1+\epsilon v_2,in{\mathbb{R}}^N,-\mathrm{\Delta }{v}_2+{\mu }_2{v}_2=(I_{\omega }\ast \mathrm{\mid }{v}_2{\mathrm{\mid }}^{2_{\omega }^{\ast }})\mathrm{\mid }{v}_2{\mathrm{\mid }}^{2_{\omega }^{\ast }-2}{v}_2+\theta q(I_{\omega }\ast \mathrm{\mid }{v}_1{\mathrm{\mid }}^p)\mathrm{\mid }{v}_2{\mathrm{\mid }}^{q-2}{v}_2+\epsilon v_1,in{\mathbb{R}}^N,{\int }_{{\mathbb{R}}^N}{v}_1^2={\alpha }_1^2,{\int }_{{\mathbb{R}}^N}{v}_2^2={\alpha }_2^2,$ where $N=3\text{or}4$, ${\alpha }_1,{\alpha }_2>0$, $\theta >0$, $2_{\omega ,\ast }:=\frac{N+\omega }{N}<p,q<2_{\omega }^{\ast }:=\frac{N+\omega }{N-2}$, $\epsilon >0$, $0<\omega <N$, $I_{\omega }:{\mathbb{R}}^N\to \mathbb{R}$ 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 $0<\theta <{\theta }_0,0<\epsilon <{\epsilon }_{\ast }$. 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 $\theta >{\theta }_{\ast },0<\epsilon <\bar{\epsilon }$.