  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.
