  We present a study of exclusive production of $\pi^{0}$ meson in
proton-proton collisions at high energies. Both diffractive bremsstrahlung
(Drell-Hiida-Deck type model), photon-photon, photon-omega and photon-odderon
exchange mechanisms are included in the calculation. The
$\pi^{0}$-bremsstrahlung contribution dominates at large (forward, backward)
pion rapidities and contributes at small $\pi^0 p$ invariant mass and could be
therefore misinterpreted as the Roper resonance $N^{*}(1440)$. Large cross
sections of the order of mb are predicted. We predict strong dependence of the
slope in $t$ (squared four-momentum transfer between ingoing and outgoing
proton) on the mass of the supplementary excited $\pi^{0} p$ system. At high
energy and midrapidity, the photon-photon contribution dominates over the
diffractive components, however, the corresponding cross section is rather
small. The photon-odderon and odderon-photon contributions are included in
addition and first estimates (upper limits) of their contributions are
presented. We suggest a search for the odderon contribution at midrapidity and
at $p_{\perp,\pi^{0}} \sim$ 0.5 GeV. Our predictions are ready for verification
at LHC. The bremsstrahlung mechanisms discussed here contribute also to the $pp
\to p(n \pi^{+})$ reaction. Both channels give a sizable contribution to the
low-mass single diffractive cross section and must be included in extrapolating
the measured experimental single diffractive cross section.
