  The cross-sections of the $\gamma p-\gamma' N'$, $\gamma p-\pi' N'$ and
$\gamma p-\gamma'\pi' N'$ reactions are calculated in the framework of the
field-theoretical one-particle ($\pi,\omega,\rho$-mesons, nucleon and
$\Delta$-resonance) exchange model. Unlike the other relativistic approaches,
our resulting amplitudes of the $\gamma p$ multichannel reactions require
one-variable covariant vertex functions as input ingredient and every diagram
of these amplitudes satisfies the current conservation condition in the Coulomb
gauge. The complete set of the model independent skeleton diagrams for the
${\gamma} p\to\gamma'\pi'N'$ reaction is presented. The separable model of the
$\pi N$ interaction is generalized to construct the spin 3/2 particle
propagator of the $\Delta$-resonance. This procedure allows to obtain the $\pi
N-\Delta$ form factor and $\Delta$ propagator directly from the $\pi N$
$P_{33}$ phase shifts. The numerical calculation of the differential cross
section of the $\gamma p-\gamma' N'$, $\gamma p-{\pi^o}' N'$ and $\gamma
p-\gamma'{\pi^o}'N'$ reactions are performed with two different separable
models of the $\Delta$ propagator and with the propagator of Breit-Wigner
shape. It is demonstrated that the numerical description of these reactions in
the $\Delta$-resonance region are very sensitive to the form of the
$\Delta$-propagator. The sensitivity of the cross-sections of the $\gamma
p\to\gamma'{\pi^o}'p'$ reaction to the magnitude of the $\Delta^+$ magnetic
moment is examined and the most convenient kinematical region for the
determination of the magnetic moment of the $\Delta^+$-resonance from the
forthcoming data is indicated.
