Geodesic
On real and ``symplectic'' meromorphic plus-matrix-function and corresponding linear fractional transformation
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 10 (2003), pp. 557-568.

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The basic result is: if linear fractional transformation with meromorphic in the unit disk nondegenerate matrix of coefficients A(z) maps the class of holomorphic contractive matrix function into itself so that real (symmetric) matrix functions are transformed into real (symmetric) matrix functions then there exists a mеromorphic scalar function ρ(z) such that ρ−1(z)A(z) is j-expansive real (“symplectic” or “antisymplectic”) matrix function. 
@article{JMAG_2003_10_a7,
     author = {L. A. Simakova},
     title = {On real and ``symplectic'' meromorphic plus-matrix-function and corresponding linear fractional transformation},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {557--568},
     publisher = {mathdoc},
     volume = {10},
     year = {2003},
     language = {ru},
     url = {https://geodesic-test.mathdoc.fr/item/JMAG_2003_10_a7/}
}
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L. A. Simakova. On real and ``symplectic'' meromorphic plus-matrix-function and corresponding linear fractional transformation. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 10 (2003), pp. 557-568. https://geodesic-test.mathdoc.fr/item/JMAG_2003_10_a7/