  In this paper we are investigated the monodromy group for linearly
polymorphic functions on compact Riemann surface of genus $g \geq 2,$ in
connection with standard uniformization of these surfaces by Kleinian groups,
and are found a neccessary and sufficients conditions, that a linearly
polymorphic function on compact Riemann surface gave a standard uniformization
of this surface. We are investigated the monodromy mapping $p : {\bf T}_{g}Q
\rightarrow \mathcal{M},$ where ${\bf T}_{g}Q$ is a vector bundle of
holomorphic quadratic abelian differentials over the Teichmueller space of
compact Riemann surfaces of genus $g,$ $\mathcal{M}$ is a space of monodromy
groups for of genus $g.$ Here is proved that over any space, which consist from
quasiconformal deformations by Koebe group of signature $\sigma = (h, s;
i_{1},..., i_{m}),$ connected with standard uniformization compact Riemann
surface of genus $g = |\sigma|,$ this mapping $p$ has the lifting of path
property . Also we are received exact variational formula for monodromy group
of the linear differential equation of the second order and the first variation
for solution of the Schwartz equation on compact Riemann surface.
