  We study the uniqueness of a vortex equation involving an entire function on
the complex plane. As geometric applications, we show that there is a unique
harmonic map $u:\mathbb{C}\rightarrow \mathbb{H}^2$ satisfying $\partial u\neq
0$ with prescribed polynomial Hopf differential; there is a unique affine
spherical immersion $u:\mathbb{C}\rightarrow \mathbb{R}^3$ with prescribed
polynomial Pick differential. We also show that the uniqueness fails for
non-polynomial entire functions with finite zeros.
