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The geometric theory of additive separation of variables is applied to thesearch for multiplicative separated solutions of the bi-Helmholtz equation. Itis shown that the equation does not admit regular separation in any coordinatesystem in any pseudo-Riemannian space. The equation is studied in the fourcoordinate systems in the Euclidean plane where the Helmholtz equation andhence the bi-Helmholtz equation is separable. It is shown that the bi-Helmoltzequation admits non-trivial non-regular separation in both Cartesian and polarcoordinates, while it possesses only trivial separability in parabolic andelliptic-hyperbolic coordinates. The results are applied to the study of smallvibrations of a thin solid circular plate of uniform density which is governedby the bi-Helmholtz equation.

Geometric theory of non-regular separation of variables and the bi-Helmholtz equation