中文摘要：

偏微分方程的精确解蕴含了方程丰富的信息，对于描述各种现象的发展规律起着至关重要的作用．因此偏微分方程的精确解成为了数学、物理、经济等领域研究的热点问题．本文研究了金融数学中最重要的模型之一Black-Scholes方程的广义分离变量解．运用条件Lie—Backlund对称与不变子空间理论相结合的方法，本文得到了形如欧拉方程的条件Lie-Backlund对称．该方程允许的条件Lie—Backlund对称与高阶变系数的常微分方程相对应．同时，我们还得到了该方程允许此特征的所有精确解．

英文摘要：

The exact solution of partial differential equations, which contains rich information for the equations, is very important for describing the development of various phenomena and thus becomes a research focus of scientific fields such as mathematics, physics, economy and so on. In this paper, the generalized separable solutions for Black-Scholes equation, which is one of most important models arising in financial mathematics, are discussed. By using the conditional Lie-Backlund symmetry and invariant subspace theory, we obtain the conditional Lie-Backlund symmetries, which are similar to Euler equation. The conditional Lie-Backlund symmetries, which are admitted by Black-Scholes, ave corresponding to high-order variable coefficient ordinary differential equations. At the same time, all of exact solutions associated to the conditional Lie-Backlund symmetries are performed.