Study of nonlinear geometric problems by methods of algebraic analysis

我们成功地用一些5阶非线性偏微分方程组完全描述了R^3中包含几个连续圆族的曲面，并显式地给出了这个方程组的非常复杂的形式。此外，作为应用，我们得到了分类这些曲面的参数个数的一个上限估计，并将这个系统归结为一个有限的一元5个未知函数的代数常微分方程组。

We succeeded in describing the surfaces in R^3 which include several continuous families of circles completely by some systems of nonlinear partial differential equations of order 5. We gave explicitly the very complicated form of this system of equations. Further, as applications, we obtained an upper estimate of the number of parameters classifying those surfaces, and a reduction of this system to a finite system of algebraic ordinary differential equations for 5 unknown functions of one variable.