Hecke symmetries associated with twisted polynomial algebras in 3 indeterminates

We consider Hecke symmetries on a 3-dimensional vector space with the associated R-symmetric algebra isomorphic to the polynomial algebra $k[x_1,x_2,x_3]$ twisted by an automorphism. The main result states that any such a Hecke symmetry is itself a twist of a Hecke symmetry with the associated R-symmetric algebra isomorphic to $k[x_1,x_2,x_3]$. This allows us to describe equivalence classes of such Hecke symmetries.