Let's consider the following differential equation.
$$\left(\frac{dy}{dx}\right)^2=(y-\alpha_1)(y-\alpha_2)(y-\alpha_3)(y-\alpha_4)$$
If we define $y=T(z)=\frac{az+b}{cz+d}$ by solving $\frac{(y-\alpha_3)(\alpha_2-\alpha_4)}{(y-\alpha_4)(\alpha_2-\alpha_3)}=\frac{(z-k^{-1})(-1+k^{-1})}{(z+k^{-1})(-1-k^{-1})}$ where k is obtained by $\frac{(k-1)^2}{(k+1)^2}=\frac{(\alpha_1-\alpha_3)(\alpha_2-\alpha_4)}{(\alpha_1-\alpha_4)(\alpha_2-\alpha_3)}$, then $T(1)=\alpha_1,T(-1)=\alpha_2,T(k^{-1})=\alpha_3,T(-k^{-1})=\alpha_4$ is satisfied, and differential equation is simplified as
$$\left(\frac{dz}{dx}\right)^2=\frac{(ad-bc)^2}{(c^2-d^2)(c^2-k^2d^2)}(1-z^2)(1-k^2 z^2),$$
and if we remove numerical factor by rescaling $x$, then it becomes the differential equations Jacobi's Elliptic function $\mathrm{sn}(x;k)$ obeys.
In general, however, $\alpha_s$ and $k$ are complex, not obeying $0<k<1$. Is there such generalization of Elliptic function? And what kind of feature does it have? (ex. relation to the Weierstrass's Elliptic function, whether to obey the differential equations)