electromagnetism - Is magnetic field of a uniformly moving charge constant everywhere?

As far as I understand, for the field of a uniformly moving charge, curl of $\mathbf E$ is zero everywhere.

Since $\nabla \times \mathbf E = -\dfrac{\partial\mathbf B}{\partial t}$, magnetic field should be constant in every point in space.

This sounds wrong, since $\mathbf B$ is supposed to fall off proportionally to $r^2$, and $r$ is changing in time for a moving charge. What is wrong with this reasoning?

Even worse, $\nabla \times \mathbf B = \dfrac{\partial\mathbf E}{\partial t}$ , and since $\dfrac{\partial\mathbf E}{\partial t}$ is not constant (because $\dfrac{\partial^2\mathbf E}{\partial t^2}$ is not zero), curl of $\mathbf B$ keeps changing.

But how can $\nabla \times \mathbf B$ keep changing if $\mathbf B$ itself stays the same?

Answer

For example, consider at $t=0$ the point charge be at the origin and moving in the $z$ direction with velocity ${\bf v}$. The electric field at this moment is $${\bf E}({\bf r})=kq\frac{1-v^2/c^2}{(1-v^2 \sin^2 \theta/c^2)^{3/2}}\frac{\hat{\bf r}}{r^2}$$ Then $$\nabla \times {\bf E}=-\frac{1}{r}\frac{\partial}{\partial \theta}kq\frac{1-v^2/c^2}{(1-v^2 \sin^2 \theta/c^2)^{3/2}}\frac{1}{r^2}\hat{{\bf \phi}}\ne {\bf 0}$$