Explicit analytic solutions for generic CP1 bifurcation thresholds

Derive explicit analytic solutions of the bifurcation equation for the critical values of the distinguished parameter at which critical points of the first kind undergo saddle-node bifurcations, for generic values of the normal-form parameters with nonzero detuning combinations.

Background

For critical points of the first kind in the inversion-symmetry case, the bifurcation thresholds are determined by the equation f_1(\mathcal E)=0, obtained from the discriminant of a fourth-degree equation in the Lagrange multiplier. The authors derive explicit formulas when \Delta_1=0, but state that no explicit analytic solutions are available for generic parameter values. Once the parameters are fixed, the equation can instead be solved numerically.