  We introduce a notion of quasi-antisymmetric Higgs $G$-bundles over curves
with marked points. They are endowed with additional structures, which replace
the parabolic structures at marked points in the parabolic Higgs bundles. The
latter means that the coadjoint orbits are attached to the marked points. The
moduli spaces of parabolic Higgs bundles are the phase spaces of complex
completely integrable systems. In our case the coadjoint orbits are replaced by
the cotangent bundles over some special symmetric spaces in such a way that the
moduli space of the modified Higgs bundles are still phase spaces of complex
completely integrable systems. We show that the moduli space of the parabolic
Higgs bundles is the symplectic quotient of the moduli space of the
quasi-antisymmetric Higgs bundle with respect to the action of product of
Cartan subgroups. Also, by changing the symmetric spaces we introduce
quasi-compact and quasi-normal Higgs bundles. Then the fixed point sets of real
involutions acting on their moduli spaces are the phase spaces of real
completely integrable systems. Several examples are given including integrable
extensions of the ${\rm SL}(2)$ Euler-Arnold top, two-body elliptic
Calogero-Moser system and the rational ${\rm SL}(2)$ Gaudin system together
with its real reductions.
