  In this paper, we consider a class of nonlocal equations where the
convolution kernel is given by a Bessel potential symbol of order $\alpha$ for
$\alpha > 1$. Based on the properties of the convolution operator, we apply a
global bifurcation technique to show the existence of a highest, even,
$2\pi$-periodic traveling-wave solution. The regularity of this wave is proved
to be exactly Lipschitz.
