An extreme solitary wave for the bidirectional Whitham system

We construct an extreme solitary wave for the bidirectional Whitham system. Its velocity profile is positive, even, continuous, smooth away from its crest and strictly decreasing to zero on the positive half-line. The crest reaches the first critical value of the cubic traveling-wave equation. The construction starts with positive highest periodic waves of arbitrarily large period. A common modulus of continuity and a fixed drop from the critical height keep their limit nonconstant. A balance identity using the kernel's first absolute moment then excludes every positive limiting pedestal, without assuming an integrable profile.