Question.Is there a more general set of equations that satisfy mean value properties, similar to the Laplacian and heat equation?

For example, finding some kernel $K(x,y)$ and a set $B(x,r)$ such that

$$u(x)=\int_{B(x,r)} u(y)K(x,y)dy.$$

Even existence of such a $K$ and a $B$ is good enough for me; eg. showing that the Fredhold integral equation has a fixed point that also solves a more general pde. Or even more generally the existence of some measure $\mu$ that gives:

$$u(x)=\int_{B(x,r)} u(y)\mu(dy).$$

Can this be approached as a fixed point problem?

For example, for $Lu=(\Delta +c)(u)$ there is an interesting MVP

$$u(\xi)=\frac{\sqrt{c}\rho}{sin(\sqrt{c}\rho)}\frac{1}{4\pi \rho^2}\int_{\partial B(\xi,\rho)} u(x)d\sigma(x).$$