On solvability of engel equations in the group of triangular matrices over a field

In the following paper we investigate commutator-type matrix equations and discuss the existence of their solutions. In particular, we derive the solutions of the Engel equations A=ek(X,Y), where ek(x,y) denotes the k-th Engel word, in the groups of unitriangular and triangular matrices over field K of arbitrary characteristic. The results directly apply to the discussion on the width of k-Engel subgroups in the considered groups. We conclude with a few observations on that matter.