The Ising correlation $C(M,N)$ for $\nu=-k$

We present Painlev{\'e} VI sigma form equations for the general Ising low and high temperature two-point correlation functions $C(M,N)$ with $M \leq N$ in the special case $\nu = -k$ where $\nu = \, \sinh 2E_h/k_BT/\sinh 2E_v/k_BT$. More specifically four different non-linear ODEs depending explicitly on the two integers $M$ and $N$ emerge: these four non-linear ODEs correspond to distinguish respectively low and high temperature, together with $M+N$ even or odd. These four different non-linear ODEs are also valid for $M \ge N$ when $\nu = -1/k$. For the low-temperature row correlation functions $C(0,N)$ with $N$ odd, we exhibit again for this selected $\nu = \, -k$ condition, a remarkable phenomenon of a Painlev\'e VI sigma function being the sum of four Painlev\'e VI sigma functions having the same Okamoto parameters. We show in this $\nu = \, -k$ case for $T < T_c$ and also $T > T_c$, that $C(M,N)$ with $M \leq N$ is given as an $N \times N$ Toeplitz determinant.