  We classify zeroth-order conservation laws of systems from the class of
two-dimensional shallow water equations with variable bottom topography using
an optimized version of the method of furcate splitting. The classification is
carried out up to equivalence generated by the equivalence group of this class.
We find additional point equivalences between some of the listed cases of
extensions of the space of zeroth-order conservation laws, which are
inequivalent up to transformations from the equivalence group. Hamiltonian
structures of systems of shallow water equations are used for relating the
classification of zeroth-order conservation laws of these systems to the
classification of their Lie symmetries. We also construct generating sets of
such conservation laws under action of Lie symmetries.
