  The one-dimensional modified shallow water equations in Lagrangian
coordinates are considered. It is shown the relationship between symmetries and
conservation laws in Lagrangian coordinates, in mass Lagrangian variables, and
Eulerian coordinates. For equations in Lagrangian coordinates an invariant
finite-difference scheme is constructed for all cases for which conservation
laws exist in the differential model. Such schemes possess the difference
analogues of the conservation laws of mass, momentum, energy, the law of center
of mass motion for horizontal, inclined and parabolic bottom topographies.
Invariant conservative difference scheme is tested numerically in comparison
with naive approximation invariant scheme.
