I'm learning about the 'PDE'
$$\nabla K(x) = \delta(x)$$
In a book it says that it is symmetric in all variables $x_1,\cdots,x_n$ and is also radially symmetric (its value depends only on $r = |x|$. Since $\delta(x) = 0$ for $x\neq 0$, then the 'PDE' requires $K$ to be harmonic for $r>0$. Thus he tries to solve the 'PDE' with a radially symmetric function $K(x)$ and end up with:
$$\frac{\partial^2 K}{\partial r^2} + \frac{n-1}{r}\frac{\partial K}{\partial r} = 0$$
Why $K$ is symmetric and radially symmetric? How did he arrive at this PDE exactly?