appears in the classical water-wave problem, as well as in the derivation of numerous dispersive equations [@MR3060183], so that our analysis might prove useful in other settings as well.

Building on the results for the integral kernel $K$ we are able to prove the main result of this paper: the existence of a highest, cusped and periodic travelling-wave solution of , monotonically increasing and smooth between its sole trough and crest in a half-period, and belonging to the Hölder space $C^{1/2}({\mathbb R})$ — but to no smaller space in the same scale. The proof thereof has two main components. The construction of a global, locally analytic, curve of sinusoidal, periodic smooth waves along which $\max\varphi \to \mu/2$ on the one hand, and a detailed analysis of solutions satisfying $\max \varphi = \mu/2$ on the other. The first part is attained via analytic global bifurcation theory developed by Buffoni, Dancer and Toland [@MR1956130], where we rule out all alternatives along the main bifurcation curve but $\max\varphi \to \mu/2$, including in particular that the curve could return to a line of constant, but nonzero, solutions (see Figure \[fig:bifurcation\] on p.  for a qualitative picture of the bifurcation diagram as a whole). It 