Nonlocally related systems, obtained through conservation law and symmetry-based methods, have proved to be useful for determining nonlocal symmetries, nonlocal conservation laws, non-invertible mappings and new exact solutions of a given partial differential equation (PDE) system. In this paper, it is shown that the symmetry-based method is a differential invariant-based method. It is shown that this allows one to naturally extend the symmetry-based method to ordinary differential equation (ODE) systems and to PDE systems with at least three independent variables. In particular, we present the situations for ODE systems, PDE systems with two independent variables and PDE systems with three or more independent variables, separately, and show that these three situations are directly connected. Examples are exhibited for each of the three situations.

通过守恒定律和基于对称性的方法获得的非局部相关系统已被证明可用于确定给定偏微分方程 (PDE) 系统的非局部对称性、非局部守恒定律、不可逆映射和新精确解。本文证明了基于对称性的方法是一种基于微分不变量的方法。结果表明，这允许将基于对称性的方法自然地扩展到常微分方程 (ODE) 系统和具有至少三个自变量的 PDE 系统。特别是，我们分别介绍了 ODE 系统、具有两个自变量的 PDE 系统和具有三个或更多自变量的 PDE 系统的情况，并表明这三种情况是直接相关的。针对三种情况中的每一种都展示了示例。