Is there some common theory behind the following two examples?
Example 1.
Let $p(t) = \sum_{n \geq 0} (-1)^k t^{2k}/(2k)!$, and $x = p(t), y = p'(t)$. Then $x^2 + y^2 = 1$ identically.
Example 2.
Fix $\omega_1, \omega_2 \in \mathbb{C}$. Let $p(t)=t^{-2}+\sum_{(m,n)\neq (0,0)} \left(\frac{1}{\left(t+m+n\omega_1/\omega_2\right)^2}-\frac{1}{\left(m+n\omega_1/\omega_2\right)^2}\right)$ be the Weierstrass rho function, and $x=p(t), y=p'(t)$. Then $y^2=4x^3-g_2x-g_3$, where $g_2,g_3$ are explicit constants depending only on $\omega_1,\omega_2$.