Graphical continuous Lyapunov models offer a novel framework for the statistical modeling of correlated multivariate data. These models define the covariance matrix through a continuous Lyapunov equation, parameterized by the drift matrix of the underlying dynamic process. In this talk, I will discuss key results on the defining equations of these models and explore the challenge of structural identifiability. Specifically, I will present conditions under which models derived from different directed acyclic graphs (DAGs) are equivalent and provide a transformational characterization of such equivalences. This is based on ongoing work with Carlos Amendola, Tobias Boege, and Ben Hollering.