  We study a nonlinear system of partial differential equations which describe
rotating shallow water with an arbitrary constant polytropic index $\gamma $
for the fluid. In our analysis we apply the theory of symmetries for
differential equations and we determine that the system of our study is
invariant under a five dimensional Lie algebra. The admitted Lie symmetries
form the $\left\{ 2A_{1}\oplus _{s}2A_{1}\right\} \oplus _{s}A_{1}$ Lie algebra
for $\gamma \neq 1$ and $2A_{1}\oplus _{s}3A_{1}$ for $\gamma =1$. The
application of the Lie symmetries is performed with the derivation of the
corresponding zero-order Lie invariants which applied to reduce the system of
partial differential equations into integrable systems of ordinary differential
equations. For all the possible reductions the algebraic or closed-form
solutions are presented. Travel-wave and scaling solutions are also determined.
