A note on the solution set of the equation $[L_1^r,P_1]=[P_2,L_2^s]$ for given linear forms $L_1,L_2$

Let $k$ be a field of characteristic zero. In this note, for given linear forms $L_1,L_2\in k[x_1,\ldots ,x_n]$ and given $r,s\in \mathbb {N}_+=\mathbb {N}\setminus \{ 0\},$ we consider the equation $[L_1^r,P_1]=[P_2,L_2^s]$ with unknowns $P_1,P_2\in k[x_1,\ldots ,x_n],$ and give a complete description of the set of all solutions of such an equation. Equivalently, the above equation can be written as an equation for differential forms: $d(L_1^r)\wedge dP_1 = dP_2\wedge d(L_2^s).$