Normalized solutions for Schrödinger systems in dimension two

In this paper, we study the existence of normalized solutions to the following nonlinear Schrödinger systems with exponential growth \begin{align*} \left{ \begin{aligned} &-Δu+λ_{1}u=H_{u}(u,v), \quad \quad \hbox{in }$\mathbb{R}$^{2},\ &-Δv+λ_{2} v=H_{v}(u,v), \quad \quad \hbox{in }$\mathbb{R}$^{2},\ &\int_{$\mathbb{R}$^{2}}|u|^{2}dx=a^{2},\quad \int_{$\mathbb{R}$^{2}}|v|^{2}dx=b^{2}, \end{aligned} \right. \end{align*} where $a,b>0$ are prescribed, ${\lambda }_1,{\lambda }_2\in \mathbb{R}$ and the functions $H_u,H_v$ are partial derivatives of a Carathéodory function $H$ with $H_u,H_v$ have exponential growth in ${\mathbb{R}}^2$. Our main results are totally new for Schrödinger systems in ${\mathbb{R}}^2$. Using the Pohozaev manifold and variational methods, we establish the existence of normalized solutions to the above problem.