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 $\rho(z)$ such that $\rho^{-1}(z) A(z)$ 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 = {http://geodesic.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. http://geodesic.mathdoc.fr/item/JMAG_2003_10_a7/