  We classified finite orbits of monodromies of the Fuchsian system for five
$2\times 2$ matrices. The explicit proof of this result is given. We have
proposed a conjecture for a similar classification for $6$ or more $2\times 2$
matrices. Cases in which all monodromy matrices have a common eigenvector are
excluded from the consideration. To classify the finite monodromies of the
Fuchsian system we combined two methods developed in this paper. The first is
an induction method: using finite orbits of smaller number of monodromy
matrices the method allows the construction of such orbits for bigger numbers
of matrices. The second method is a formalism for representing the tuple of
monodromy matrices in a way that is invariant under common conjugation way,
this transforms the problem into a form that allows one to work with rational
numbers only.
  The classification developed in this paper can be considered as the first
step to a classification of algebraic solutions of the Garnier system.
