  The geometric theory of additive separation of variables is applied to the
search for multiplicative separated solutions of the bi-Helmholtz equation. It
is shown that the equation does not admit regular separation in any coordinate
system in any pseudo-Riemannian space. The equation is studied in the four
coordinate systems in the Euclidean plane where the Helmholtz equation and
hence the bi-Helmholtz equation is separable. It is shown that the bi-Helmoltz
equation admits non-trivial non-regular separation in both Cartesian and polar
coordinates, while it possesses only trivial separability in parabolic and
elliptic-hyperbolic coordinates. The results are applied to the study of small
vibrations of a thin solid circular plate of uniform density which is governed
by the bi-Helmholtz equation.
