  The geometric model of the pathway linking Schlesinger and
Garnier--Painlev\'e~6 systems based on an original orthonormalization of a set
of elements in ${sl(2,\mathbb C)}$ is constructed. The explicit polynomial map
of the Cartesian products of $n-2$ quadrics (the Zariski-topology chart of the
phase space of the Garnier--Painlev\'e~6 system) into the phase space of the
Schlesinger system and the rational inverse to this map are presented.
