Artin-Schelter regular algebras of dimension five with two generators

We study and classify Artin-Schelter regular algebras of dimension five with two generators under an additional ${\mathbb{Z}}^2$-grading by Hilbert driven Gr"{o}bner basis computations. All the algebras we obtained are strongly noetherian, Auslander regular, and Cohen-Macaulay. One of the results provides an answer to Fl{\o}ystad-Vatne’s question in the context of $\mathbb Z^2$-grading. Our results also achieve a connection between Lyndon words and Artin-Schelter regular algebras.