  We introduce the notion of a real form of a Hamiltonian dynamical system in
analogy with the notion of real forms for simple Lie algebras. This is done by
restricting the complexified initial dynamical system to the fixed point set of
a given involution. The resulting subspace is isomorphic (but not
symplectomorphic) to the initial phase space. Thus to each real Hamiltonian
system we are able to associate another nonequivalent (real) ones. A crucial
role in this construction is played by the assumed analyticity and the
invariance of the Hamiltonian under the involution. We show that if the initial
system is Liouville integrable, then its complexification and its real forms
will be integrable again and this provides a method of finding new integrable
systems starting from known ones. We demonstrate our construction by finding
real forms of dynamics for the Toda chain and a family of Calogero--Moser
models. For these models we also show that the involution of the complexified
phase space induces a Cartan-like involution of their Lax representations.
