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This paper is dedicated to studying matrix solutions of the cubic Szeg\H{o}equation on the line in Pocovnicu [arXiv:1001.4037, arXiv:1012.2943] andG\'erard--Pushnitski [arXiv:2307.06734], leading to the following matrixSzeg\H{o} equation on $\mathbb{R}$, \begin{equation*} i \partial_t U =\Pi_{\geq 0} \left(U U ^* U \right), \quad \widehat{\left(\Pi_{\geq 0}U\right)}(\xi )= \mathbf{1}_{\xi \geq 0}\hat{U}(\xi )\in \mathbb{C}^{M \timesN}. \end{equation*} Inspired from the space-periodic matrix Szeg\H{o} equationin Sun [arXiv:2309.12136], we establish its Lax pair structure via doubleHankel operators and Toeplitz operators. Then the explicit formula in[arXiv:2307.06734] can be extended to two equivalent formulas in the matrixequation case, which both express every solution explicitly in terms of itsinitial datum and the time variable.

Matrix solutions of the cubic Szeg\H{o} equation on the real line