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We study a nonlinear system of partial differential equations that describe rotating shallow water with an arbitrary constant polytropic index  γ  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{    2    A  1⊕  s     2   A  1}⊕  sA  1$\left\{{2{A_{1}}{\ \oplus_{s}}\ 2{A_{1}}}\right\}{\ \oplus_{s}}\ {A_{1}}$    Lie algebra for  γ  ≠ 1 and2    A  1⊕  s     3   A  1$2{A_{1}}{\ \oplus_{s}}\ 3{A_{1}}$    for  γ  = 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.

Lie Symmetries and Similarity Solutions for Rotating Shallow Water