  The sigma form of the Painlev{\'e} VI equation contains four arbitrary
parameters and generically the solutions can be said to be genuinely
``nonlinear'' because they do not satisfy linear differential equations of
finite order. However, when there are certain restrictions on the four
parameters there exist one parameter families of solutions which do satisfy
(Fuchsian) differential equations of finite order. We here study this phenomena
of Fuchsian solutions to the Painlev{\'e} equation with a focus on the
particular PVI equation which is satisfied by the diagonal correlation function
C(N,N) of the Ising model. We obtain Fuchsian equations of order $N+1$ for
C(N,N) and show that the equation for C(N,N) is equivalent to the $N^{th}$
symmetric power of the equation for the elliptic integral $E$.
  We show that these Fuchsian equations correspond to rational algebraic curves
with an additional Riccati structure and we show that the Malmquist Hamiltonian
$p,q$ variables are rational functions in complete elliptic integrals. Fuchsian
equations for off diagonal correlations $C(N,M)$ are given which extend our
considerations to discrete generalizations of Painlev{\'e}.
