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 $${\bf E}({\bf r})=k\frac{1-v^2/c^2}{(1-v^2 \sin^2 \theta/c^2)^{3/2}}\frac{\hat{\bf r}}{r^2}$$$${\bf E}({\bf r})=kq\frac{1-v^2/c^2}{(1-v^2 \sin^2 \theta/c^2)^{3/2}}\frac{\hat{\bf r}}{r^2}$$ Then $$\nabla \times {\bf E}=-\frac{1}{r}\frac{\partial}{\partial \theta}k\frac{1-v^2/c^2}{(1-v^2 \sin^2 \theta/c^2)^{3/2}}\frac{1}{r^2}\hat{{\bf \phi}}\ne {\bf 0}$$$$\nabla \times {\bf E}=-\frac{1}{r}\frac{\partial}{\partial \theta}kq\frac{1-v^2/c^2}{(1-v^2 \sin^2 \theta/c^2)^{3/2}}\frac{1}{r^2}\hat{{\bf \phi}}\ne {\bf 0}$$
