  In this paper we discuss integrable higher order equations {\em of
Camassa-Holm (CH) type}. Our higher order CH-type equations are "geometrically
integrable", that is, they describe one-parametric families of pseudo-spherical
surfaces, in a sense explained in Section 1, and they are integrable in the
sense of zero curvature formulation ($\simeq$ Lax pair) with infinitely many
local conservation laws. The major focus of the present paper is on a specific
fifth order CH-type equation admitting {\em pseudo-peakons} solutions, that is,
weak bounded solutions with differentiable first derivative and continuous and
bounded second derivative, but such that any higher order derivative blows up.
Furthermore, we investigate the Cauchy problem of this fifth order CH-type
equation on the real line and prove local well-posedness under the initial
conditions $u_0 \in H^s(\mathbb{R})$, $s > 7/2$. In addition, we study
conditions for global well-posedness in $H^4(\mathbb{R})$ as well as conditions
causing local solutions to blow up in a finite time. We conclude our paper with
some comments on the geometric content of the high order CH-type equations.
