在有限维可积Hamilton系统的研究中，Hamilton-Jacobi方程的可分性是一个经典而又活跃的研究课题。可分性与微分几何有着密切的联系。对于欧氏平面上的Hamilton系统，人们已有判别其可分性的经典Bertrand-Darboux(BD)定理。同时一些非欧氏平面上的系统也有很多研究结果。然而现有文献有不足之处，没有探讨这些空间上的类BD定理。在前人研究的基础上，我们拟从Bertrand-Darboux定理的角度来研究这些非欧氏空间上的Hamilton系统。对于一些常见的非欧氏空间，例如球面、双曲平面、Minkowski平面等，我们拟给出类BD定理(方程)的具体形式，并试图初步揭示出这一定理与文献中已有结果之间的内在联系。这些研究能进一步揭示出经典的可积Hamilton系统理论与现代微分几何之间的密切关系，对于加深数学与物理不同分支之间的渗透和交叉有着重要的意义。

The separability of Hamilton-Jacobi equation is a classical but active topic among the research of classical integrable Hamiltonian system. It is intimately related with differential geometry. For the system defined on Euclidean plane, the Bertrand-Darboux (BD) theorem is a classical tool to detect its separability. Systems defined on non-Euclidean spaces are also studied. However, the similar BD-type theorems on them have not been discussed. We will investigate Hamiltonian systems defined on non-Euclidean spaces, from the perspective of BD-type theorem. More precisely, we will establish the explicit BD-type theorems on some common spaces, e.g. the sphere, hyperbolic plane, Minkowski plane; the relation between BD-type theorems and some known results will be discussed. These study may reveal the close connection between the theory of classical integable system and modern differential geometry, and plays a vital role in the interdisciplinary investigation in mathematical physics.