  We construct algebro-geometric upper triangular solutions of rank two
Schlesinger systems. Using these solutions we derive two families of solutions
to the sixth Painlev\'e equation with parameters $({1}/{8}, -{1}/{8}, {1}/{8},
{3}/{8})$ expressed in simple forms using periods of differentials on elliptic
curves. Similarly for every integer $n$ different from $0$ and $-1$ we obtain
one family of solutions to the sixth Painlev\'e equation with parameters
$(\frac{9n^2+12n+4}{8}, -\frac{n^2}{8}, \frac{n^2}{8}, \frac{4-n^2}{8})$.
