  Analytic continuation of the classical dynamics generated by a standard
Hamiltonian, H = p^2/2m + v(x), into the complex plane yields a particular
complex classical dynamical system. For an analytic potential v, we show that
the resulting complex system admits a description in terms of the phase space
R^4 equipped with an unconventional symplectic structure. This in turn allows
for the construction of an equivalent real description that is based on the
conventional symplectic structure on R^4, and establishes the equivalence of
the complex extension of classical mechanics that is based on the
above-mentioned analytic continuation with the conventional classical
mechanics. The equivalent real Hamiltonian turns out to be twice the real part
of H, while the imaginary part of H plays the role of an independent integral
of motion ensuring the integrability of the system. The equivalent real
description proposed here is the classical analog of the equivalent Hermitian
description of unitary quantum systems defined by complex, typically
PT-symmetric, potentials.
