  It is very well known that periodic orbits of autonomous Hamiltonian systems
are generically organized into smooth one-parameter families (the parameter
being just the energy value). We present a simple example of an integrable
Hamiltonian system (with an arbitrary number of degrees of freedom greater than
one) with a unique periodic orbit in the phase space (which is not compact).
Similar examples are given for Hamiltonian systems with a unique invariant
torus (of any prescribed dimension) carrying conditionally periodic motions.
Parallel examples for Hamiltonian systems with a compact phase space and with
uniqueness replaced by isolatedness are also constructed. Finally, reversible
analogues of all the examples are described.
