Geodesic
The structure of solutions of the matrix equation $J_n(0) Y + Y^{\mathsf{T}} J_n(0) = 0$ for even $n$
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXIII, Tome 496 (2020), pp. 97-100 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is shown that every solution of the matrix equation in the title of the paper, where $n = 2m$, can be transformed by a symmetric permutation of rows and columns to the direct sum of two triangular Toeplitz matrices of order $m$. 
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     title = {The structure of solutions of the matrix equation $J_n(0) Y + Y^{\mathsf{T}} J_n(0) = 0$ for even $n$},
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Kh. D. Ikramov. The structure of solutions of the matrix equation $J_n(0) Y + Y^{\mathsf{T}} J_n(0) = 0$ for even $n$. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXIII, Tome 496 (2020), pp. 97-100. http://geodesic.mathdoc.fr/item/ZNSL_2020_496_a6/

[1] F. De Terán, F. M. Dopico, “The solution of the equation $XA + AX^T = 0$ and its application to the theory of orbits”, Linear Algebra Appl., 434 (2011), 44–67 | DOI | MR | Zbl

[2] V. V. Voevodin, E. E. Tyrtyshnikov, Vychislitelnye protsessy s teplitsevymi matritsami, Nauka, M., 1987 | MR