We are concerned with a critical Choquard system with prescribed mass
\(\begin{cases}-\Delta u+\lambda_{1}u=(I_{\mu}\ast\vert u\vert^{2_{\mu}^{\ast}})\vert u\vert^{2_{\mu}^{\ast}-2}u+\nu p(I_{\mu}\ast\vert v\vert^{q})\vert u\vert^{p-2}u & \text{in}\, \mathbb{R}^{N},\\-\Delta v+\lambda_{2}v=(I_{\mu}\ast\vert v\vert^{2_{\mu}^{\ast}})\vert v\vert^{2_{\mu}^{\ast}-2}v+\nu q(I_{\mu}\ast\vert u\vert^{q})\vert v\vert^{p-2}v & \text{in}\, \mathbb{R}^{N},\\\int_{\mathbb{R}^{N}}u^{2}=a^{2},\qquad\int_{\mathbb{R}^{N}}v^{2}=b^{2}, \end{cases}\)
where N ≥ 3, 0 < μ < N, ν ∈ ℝ, Iμ: ℝN → ℝ is a Riesz potential, \(2_{\mu}^{\ast}:={2N-\mu\over{N-2}}\) and \({2N-\mu\over{N-2}}<p,q<2_{\mu}^{\ast}\) . In L2-subcritical, L2-critical and L2-supercritical cases, we obtain the existence, nonexistence and limiting profiles of normalized solutions. In particular, we reveal the relation between the existence of normalized solutions and the parameter μ. Meanwhile, in the L2-subcritical case, the system admits a second normalized solution which is of mountain-pass type when N ∈ {3, 4} and a normalized ground state which is a local minimizer when N ≥ 5.
