This is another one of those examples where textbooks always just gloss over it with the remark that it "can be done" and then just state the result.
I want to compute the general form of a van Hove singularity if the dispersion relation expanded to second order has a saddle point. As a simple example, consider
$$E = E_0 + a_x k_x^2 - a_y k_y^2$$ where the coefficients $a_x$ and $a_y$ are both positive.
I understand how the derivation works when all coefficients are negative or all coefficients are positive, because then the surface of constant energy is actually finite: It's an ellipsoid and we can estimate its volume.
But how to proceed for saddle point?
In the 2D-example above, I have $$D(E) \propto \int_{E(k_x,k_y) = E} \frac{1}{\sqrt{a_x^2 k_x^2 + a_y^2 k_y^2}}$$
and I'm not sure how to go from there. Do I actually find a parametrization for that path and compute that integral or is there a better way to arrive at the approximate form?