A new universal real flow of the Hilbert-cubical type

We provide a new universal real flow of the Hilbert-cubical type. We prove that any real flow can be equivariantly embedded in the translation on $L(\mathbb{R})^\mathbb{N}$, where $L(\mathbb{R})$ denotes the space of $1$-Lipschitz functions $f:\mathbb{R}\to[0,1]$. Furthermore, all those functions in $L(\mathbb{R})^\mathbb{N}$ that are images of such embeddings can be chosen as $C^1$-functions.