^3 \left(\rho ^2+1\right) \cos \phi }{\left(1-\rho
   ^2 \right)^2 \left(1+ \rho ^2 \cos 2 \phi \right)}\end{aligned}$$ where we allow $\rho$ and $\phi$ to sweep over their full ranges $0 \le \rho \le1$, $-\pi \le \phi \le \pi$.

This parametrization provides some interesting information. For instance, it shows that $K$ increases monotonically with $\rho$, for each fixed value of $\phi$. Hence $K \ge  2$ everywhere on the saddle-node surface, with the minimum value $K=2$ being attained when $\rho=0$ and hence $F=0$, or in other words, when there is no forcing. This result makes sense. In the absence of forcing, the system is just the traditional Kuramoto model with a Lorentzian $g(\omega)$, and $K = 2 \Delta$ (or in dimensionless terms, $K=2$) is the well-known formula for the critical coupling at the onset of mutual synchronization \[Kuramoto 1975, 1984\].

To compare our results with those obtained numerically by Antonsen et al. \[2008\], it is more illuminating to slice through the saddle-node surface at a fixed value of $K>2$ and then plot the resulting saddle-node curves in the $(\Omega, F)$ plane. To find these curves we solve $\rho' = 0$, $\phi' = 0$ and $\delta=0$ for $\Omega, \sin \phi$ and $\cos \phi$, and then 