  The germ of the universal isomonodromic deformation of a logarithmic
connection on a stable n-pointed genus g curve always exists in the analytic
category. The first part of this paper investigates under which conditions it
is the analytic germification of an algebraic isomonodromic deformation. Up to
some minor technical conditions, this turns out to be the case if and only if
the monodromy of the connection has finite orbit under the action of the
mapping class group. The second part of this paper studies the dynamics of this
action in the particular case of reducible rank 2 representations and genus g >
0, allowing to classify all finite orbits. Both of these results extend recent
ones concerning the genus 0 case.
