Non-stationary Lamé equation and WZW/Virasoro conformal blocks

The non-stationary Lamé equation

$$\left(2i \pi \kappa \partial_\tau -\partial_x^2+g(g-1)\wp(x)\right)\psi(x,\tau)=E \psi(x,\tau)$$

appears as a BPZ type equation (for the 2 points Virasoro conformal blocks with one degenerate field on the torus), and as a KZ type equation (for the one point conformal blocks on the torus of a WZW model).

1) I am not familiar with the WZW model. Why does this equation appear twice? Is it just a coincidence?

2) Do these conformal blocks give different kind of solution to this equation?