The Ising correlation $C(M,N)$ for $ u=-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 $ u = -k$ where $ u = , 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 $ u = -1/k$. For the low-temperature row correlation functions $ C(0,N)$ with $ N$ odd, we exhibit again for this selected $ u = , -k$ condition, a remarkable phenomenon of a Painleve VI sigma function being the sum of four Painleve VI sigma functions having the same Okamoto parameters. We show in this $ u = , -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.