We obtained constructive necessary and sufficient conditions for the solvability and a scheme for constructing solutions of a linear matrix equation in the form of structured matrices. In particular case we obtained constructive necessary and sufficient conditions for the solvability and a scheme for constructing solutions of a linear matrix equation with an $\mathcal{L}$-structure. Particular instances of the $\mathcal{L}$-structure are magic squares, Hilbert, Hankel and Toeplitz matrices, Hermitian, symmetric and skew-symmetric matrices, as well as quaternions and biquaternions.

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