 Saddle-point optical transition contour

For a local transition-energy model $E_c-E_v=E_s+A(k_x^2-k_y^2)$ with $A>0$, positive detuning has two hyperbola branches opening along x. Negative detuning opens along y; the zero-detuning limiting contour is a pair of intersecting lines. The coarea integrand on one positive-detuning branch reduces to $d{k}_y/[2A\sqrt{k_y^2+\delta /A}]$. The gradient vanishes at the saddle, so this is a limiting description as nonzero detuning approaches zero.