  In 2003, A.J. Majda and J.A. Biello derived and studied the so-called reduced
equations for equatorial baroclinic-barotropic waves, to which we refer as to
the Majda-Biello system. The equations in question describe the nonlinear
interaction of long-wavelength equatorial Rossby waves and barotropic Rossby
waves with a significant midlatitude projection in the presence of suitable
horizontally and vertically sheared zonal mean flows. Below we present a
Hamiltonian structure for Majda-Biello system and describe all generalized
symmetries and conservation laws for the latter. It turns out that there are
only three symmetries corresponding to x-translations, t- translations and to a
scaling of t, x, u and v, and four conservation laws, one of which is
associated to the conservation of energy, the second conserved quantity is just
the Hamiltonian functional and the other two are Casimir functionals of the
Hamiltonian operator admitted by our system. Our result provides inter alia a
rigorous proof of the fact that the Majda-Biello system has just the
conservation laws mentioned in the paper by Majda and Biello.
