Koopman Embedding and Super-Linearization Counterexamples with Isolated Equilibria

A frequently repeated claim in the "applied Koopman operator theory'' literature is that a dynamical system with multiple isolated equilibria cannot be linearized in the sense of admitting a smooth embedding as an invariant submanifold of a linear dynamical system. This claim is sometimes made only for the class of super-linearizations, which additionally require that the embedding "contain the state''. We show that both versions of this claim are false by constructing (super-)linearizable smooth dynamical systems on $\mathbb{R}^k$ having any countable (finite) number of isolated equilibria for each $k>1$.