Differentiation Equation Curl, Divergence

How do I solve these:

$\displaystyle \nabla \cdot \mathbf{A} = 0$

$\displaystyle \nabla \times \mathbf{A} = 0$

Where $\displaystyle \mathbf{A} =\left(

\begin{array}{c}

A_x \\

A_y \\

A_z

\end{array}

\right) $

From the first equation:

$\displaystyle \dfrac{\partial A_x}{\partial x}+\dfrac{\partial A_y}{\partial y}+\dfrac{\partial A_z}{\partial z}=0$

From the second equation:

$\displaystyle \dfrac{\partial A_x}{\partial y}=\dfrac{\partial A_y}{\partial x}$

$\displaystyle \dfrac{\partial A_x}{\partial z}=\dfrac{\partial A_z}{\partial x}$

$\displaystyle \dfrac{\partial A_y}{\partial z}=\dfrac{\partial A_z}{\partial y}$

What next?