On pairs of commuting derivations of the polynomial ring in one or two variables

It is well known that each pair of commuting linear operators on a finite dimensional vector space over an algebraically closed field has a common eigenvector. We prove an analogous statement for derivations of k[x]k[x] and k[x,y]k[x,y] over any field k   of zero characteristic. In particular, if D1D1 and D2D2 are commuting derivations of k[x,y]k[x,y] and they are linearly independent over k  , then either (i) they have a common polynomial eigenfunction; i.e., a nonconstant polynomial f∈k[x,y]f∈k[x,y] such that D1(f)=λfD1(f)=λf and D2(f)=μfD2(f)=μf for some λ,μ∈k[x,y]λ,μ∈k[x,y], or (ii) they are Jacobian derivationsDu(g):=∂u∂x∂u∂y∂g∂x∂g∂y,Dv(g):=∂v∂x∂v∂y∂g∂x∂g∂yforallg∈k[x,y]defined by some u,v∈k[x,y]u,v∈k[x,y] for which Du(v)Du(v) is a nonzero constant.