Geodesic
Solvability conditions for the matrix equation $X^{\mathrm{T}}DX+AX+X^{\mathrm{T}}B+C=0$
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 4, pp. 555-557

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Under certain restrictions on the matrix coefficients of the quadratic matrix equation $X^{\mathrm{T}}DX+AX+X^{\mathrm{T}}B+C=0$, solvability conditions for this equation are given, and its relationship with an equation of the form $XAX=B$ is found. Certain specific types of matrix coefficients obeying the above solvability conditions are indicated. 
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     author = {Yu. O. Vorontsov},
     title = {Solvability conditions for the matrix equation $X^{\mathrm{T}}DX+AX+X^{\mathrm{T}}B+C=0$},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {555--557},
     publisher = {mathdoc},
     volume = {55},
     number = {4},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_4_a2/}
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Yu. O. Vorontsov. Solvability conditions for the matrix equation $X^{\mathrm{T}}DX+AX+X^{\mathrm{T}}B+C=0$. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 4, pp. 555-557. http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_4_a2/