A note on the solution set of the equation [Lr1, P1]=[P2, Ls2] for given linear forms L1, L2

Let k be a field of characteristic zero. In this note, for given linear forms L1, L2 ∈ k[x1,…, xn] and given r, s ∈ N+ = N \ {0}, we consider the equation [Lr1, P1] =[P2, Ls2] with unknowns P1, P2 ∈ k[x1, . . . , xn], and give a complete description of the set of all solutions of such an equation. Equivalently, the above equation can be written as anequation for differential forms: d(Lr1) ∧ dP1 = dP2 ∧ d(Ls2).