This paper presents a method for analyzing the dynamics of large systems of globally coupled phase oscillators, showing that their behavior can be described by a finite set of nonlinear ordinary differential equations in the infinite size limit. The authors demonstrate that for certain systems, such as the Kuramoto model with a Lorentzian frequency distribution, the macroscopic evolution can be reduced to a low-dimensional description. This approach is applicable to various extensions of the Kuramoto model, including systems with external driving and communities of oscillators, as well as systems with time-delayed coupling. The key idea is to consider a restricted class of states that allows for an exact reduction of the system's dynamics to a finite set of equations. This reduction is achieved by defining a specific form for the distribution function of oscillator phases, which leads to a set of nonlinear ordinary differential equations that describe the macroscopic behavior of the system. The authors show that this approach yields results consistent with those obtained from more general methods, suggesting its broad applicability. For the Kuramoto model with a Lorentzian frequency distribution, the authors derive an exact solution for the nonlinear time evolution of the system. They find that the order parameter, which quantifies the degree of synchronization, evolves according to a single nonlinear ordinary differential equation. This equation captures the essential dynamics of the system, including the transition from incoherent to coherent behavior as the coupling strength increases. The method is also extended to other cases, such as systems with external driving, communities of oscillators, and time-delayed coupling. In each case, the authors show that the dynamics can be reduced to a finite set of equations, demonstrating the versatility of the approach. The results suggest that the low-dimensional description is not only accurate but also useful for understanding the behavior of complex systems with many interacting oscillators. The paper concludes that this method provides a powerful tool for analyzing the dynamics of large systems of coupled oscillators.This paper presents a method for analyzing the dynamics of large systems of globally coupled phase oscillators, showing that their behavior can be described by a finite set of nonlinear ordinary differential equations in the infinite size limit. The authors demonstrate that for certain systems, such as the Kuramoto model with a Lorentzian frequency distribution, the macroscopic evolution can be reduced to a low-dimensional description. This approach is applicable to various extensions of the Kuramoto model, including systems with external driving and communities of oscillators, as well as systems with time-delayed coupling. The key idea is to consider a restricted class of states that allows for an exact reduction of the system's dynamics to a finite set of equations. This reduction is achieved by defining a specific form for the distribution function of oscillator phases, which leads to a set of nonlinear ordinary differential equations that describe the macroscopic behavior of the system. The authors show that this approach yields results consistent with those obtained from more general methods, suggesting its broad applicability. For the Kuramoto model with a Lorentzian frequency distribution, the authors derive an exact solution for the nonlinear time evolution of the system. They find that the order parameter, which quantifies the degree of synchronization, evolves according to a single nonlinear ordinary differential equation. This equation captures the essential dynamics of the system, including the transition from incoherent to coherent behavior as the coupling strength increases. The method is also extended to other cases, such as systems with external driving, communities of oscillators, and time-delayed coupling. In each case, the authors show that the dynamics can be reduced to a finite set of equations, demonstrating the versatility of the approach. The results suggest that the low-dimensional description is not only accurate but also useful for understanding the behavior of complex systems with many interacting oscillators. The paper concludes that this method provides a powerful tool for analyzing the dynamics of large systems of coupled oscillators.