On maximal subalgebras

Let k be an algebraically closed field. We classify all maximal k-subalgebras of k[t,t−1,y]. To the authors' knowledge, this is the first such classification result for a commutative algebra of dimension >1. Moreover, we classify all maximal k-subalgebras of k[t,y] that contain a coordinate of k[t,y]. Furthermore, we give plenty examples of maximal k-subalgebras of k[t,y] that do not contain a coordinate.