On linear equations in free Lie algebras

We investigate equations of the form [x1,u1]+⋯+[xk,uk]=0 over a free Lie algebra L. In the case where the coefficients u1,…,uk are free generators of L, we generalize a number of earlier results on equations with two variables to equations with an arbitrary number of indeterminates. Our main results refer to the case where the coefficients coincide with the free generators of L. We give a detailed description of the solution space and we obtain an explicit basis for its multilinear fine homogeneous component.