  In [9], the celebrated K{\L}-inequality has been extended from definable
functions $f:\mathbb{R}^{n}\rightarrow\mathbb{R} $ to definable multivalued
maps $S:\mathbb{R}\rightrightarrows\mathbb{R}^{n}$, by establishing that the
co-derivative mapping $D^{\ast}S$ admits a desingularization around every
critical value. As was the case in the gradient dynamics, this
desingularization yields a uniform control of the lengths of all bounded orbits
of the corresponding sweeping process $-\dot{\gamma}(t)\in
N_{S(t)}(\gamma(t))$. In this paper, working outside the framework of o-minimal
geometry, we characterize the existence of a desingularization for the
coderivative in terms of the behaviour of the sweeping process orbits and the
integrability of the talweg function. These results are close in spirit with
the ones in [3], where characterizations for the desingularization of the
(sub)gradient of functions had been obtained.
