In the spirit of optimal approximation and reduced order modeling, the goal of DMD methods and variants is to describe the dynamical evolution as a linear evolution in an appropriately transformed lower rank space, as best as possible. First we will review some differences between the Koopman operator and the Frobenius-Perron operator in its dual space. Then lead to noticing that Koopman eigenfunctions follow a linear PDE that is solvable by the method of characteristics yields several interesting relationships between geometric and algebraic properties. We focus on contrasting cardinality, algebraic multiplicity and other geometric aspects with the introduction of an equivalence class, “primary eigenfunctions,” for those eigenfunctions with identical sets of level sets. We present a construction that leads to functions on the data surface that yield optimal Koopman eigenfunction DMD, (oKEEDMD). We will also describe that disparate systems can be “matched” transformed by a diffeomorphism constructed via eigenfunctions from each system, a reinterpretation of integrability, computationally stated by our “matching extended dynamic mode decomposition (EDMD)” (EDMD-M).