It is shown that the nonlinear system of equations, describing spatial fluctuations of the multilayered stratified shallow liquid in a rotating circular parabolic basin, can be transformed to the classical multilayered shallow water equations. This transformation is obtained as a result of the analysis of symmetry properties of the equations of motion of a rotating liquid and a more general model considering piecewise-constant stratification of a liquid. The existence of the not trivial transformations of the equations of motion has allowed generating new solutions. A new class of periodic solutions, which describes nonlinear fluctuations of the multilayered stratified liquid in a circular paraboloid with closed or ergodic trajectories of liquid particles, is obtained and studied.
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