A new way to use nonlocal symmetries to determine first integrals of second-order nonlinear ordinary differential equations

Finding first integrals of second-order nonlinear ordinary differential equations (nonlinear 2ODEs) is a very difficult task. In very complicated cases, where we cannot find Darboux polynomials (to construct an integrating factor) or a Lie symmetry (that allows us to simplify the equations), we sometimes can solve the problem by using a nonlocal symmetry. In [1], [2], [3] we developed (and improved) a method (S-function method) that is successful in finding nonlocal Lie symmetries to a large class of nonlinear rational 2ODEs. However, even with the nonlocal symmetry, we still need to solve a 1ODE (which can be very difficult to solve) to find the first integral. In this work we present a novel way of using the nonlocal symmetry to compute the first integral with a very efficient linear procedure.求解二阶非线性常微分方程（nonlinear 2ODEs）的首次积分是一项极具难度的任务。在极为复杂的场景中，若无法找到达布多项式（Darboux polynomials）以构造积分因子（integrating factor），亦或无法获取李对称（Lie symmetry）来对方程进行简化，我们有时可借助非局部对称（nonlocal symmetry）解决该问题。在文献[1]、[2]、[3]中，我们提出并改进了一种S函数法（S-function method），该方法可有效针对大类非线性有理2ODEs求解非局部李对称。然而，即便借助非局部对称，我们仍需求解一个一阶常微分方程（1ODE）——该方程的求解往往难度极高——才能获得首次积分。本研究提出了一种全新方法，可借助非局部对称，通过高效的线性流程计算首次积分。