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.

中文翻译：

---

非正则变量分离的几何理论和双亥姆霍兹方程

变量加法分离的几何理论被应用于搜索双亥姆霍兹方程的乘法分离解。结果表明，该方程不允许在任何伪黎曼空间中的任何坐标系中进行规则分离。该方程是在欧几里得平面的四个坐标系中研究的，其中亥姆霍兹方程和双亥姆霍兹方程是可分离的。结果表明，bi-Helmoltz 方程在笛卡尔坐标和极坐标中都存在非平凡的非正则分离，而在抛物线和椭圆-双曲线坐标中仅具有平凡的可分离性。结果应用于研究由双亥姆霍兹方程控制的均匀密度的薄实心圆板的小振动。