  We study the factorizations of Ising low-temperature correlations C(M,N) for
$\nu=-k$ and M+N odd, $M \le N$, for both the cases $M\neq 0$ where there are
two factors, and $M=0$ where there are four factors. We find that the two
factors for $ M \neq 0$ satisfy the same non-linear differential equation and,
similarly, for M=0 the four factors each satisfy Okamoto sigma-form of
Painlev\'e VI equations with the same Okamoto parameters. Using a Landen
transformation we show, for $M\neq 0$, that the previous non-linear
differential equation can actually be reduced to an Okamoto sigma-form of
Painlev\'e VI equation. For both the two and four factor case, we find that
there is a one parameter family of boundary conditions on the Okamoto
sigma-form of Painlev\'e VI equations which generalizes the factorization of
the correlations C(M,N) to an additive decomposition of the corresponding
sigma's solutions of the Okamoto sigma-form of Painlev\'e VI equation which we
call lambda extensions. At a special value of the parameter, the
lambda-extensions of the factors of C(M,N) reduce to homogeneous polynomials in
the complete elliptic functions of the first and second kind. We also
generalize some Tracy-Widom (Painlev\'e V) relations between the sum and
difference of sigma's to this Painlev\'e VI framework.
