As we know that the maxwells equation in free space are  

$$\nabla\cdot \vec{E}=0$$
$$\nabla\cdot \vec{B}=0$$
$$\nabla\times \vec{E}=-\frac{\partial \vec{B}}{\partial t}$$
$$\nabla \times \vec{B}= \frac{1}{c^2}\frac{\partial \vec{E}}{\partial t}$$  

Now if we take $\nabla \times \vec{E}=0$, that is if $ \frac{\partial \vec{B}}{\partial t}=0 $, will the $ \frac{\partial \vec{E}}{\partial t} $ be non zero. From here we can see that it can not represent a em wave equation. That's why I am curuius it will lead to $ \frac{\partial \vec{E}}{\partial t}=0 $ or it will still remain non zero. I have tried several ways to solve the equation taking $\nabla \times \vec{E}=0$ in the third equation but could not arrive at any conclution.