Implicit Runge–Kutta (IRK) methods are highly effective for solving stiff ordinary differential equations (ODEs) but can be computationally expensive for largescale problems due to the need of solving coupled algebraic equations at each step. The key idea here is to reformulate a perturbed stage system in a stable way and to retrieve the exact solution through the solution of a Sylvester matrix equation with a known low-rank structure on the right-hand side. We focus on two major IRK families—symmetric and collocation schemes—and extend the methodology to nonlinear settings via a simplified Newton iteration. A set of numerical experiments, including ODEs derived from spatial discretizations of PDEs, confirm the effectiveness of the proposed approach.