  We start with a review of a class of systems with invariant relations, so
called {\it systems of Hess--Appel'rot type} that generalizes the classical
Hess--Appel'rot rigid body case. The systems of Hess-Appel'rot type carry an
interesting combination of both integrable and non-integrable properties.
Further, following integrable line, we study partial reductions and systems
having what we call the {\it Zhukovskii property}: these are Hamiltonian
systems with invariant relations, such that partially reduced systems are
completely integrable. We prove that the Zhukovskii property is a quite general
characteristic of systems of Hess-Appel'rote type. The partial reduction
neglects the most interesting and challenging part of the dynamics of the
systems of Hess-Appel'rot type - the non-integrable part, some analysis of
which may be seen as a reconstruction problem. We show that an integrable
system, the magnetic pendulum on the oriented Grassmannian $Gr^+(4,2)$ has
natural interpretation within Zhukovskii property and it is equivalent to a
partial reduction of certain system of Hess-Appel'rot type. We perform a
classical and an algebro-geometric integration of the system, as an example of
an isoholomorphic system. The paper presents a lot of examples of systems of
Hess-Appel'rot type, giving an additional argument in favor of further study of
this class of systems.
