Generalised Lyndon-Schützenberger Equations

We fully characterise the solutions of the generalised Lyndon-Schutzenberger word equations u 1 ⋯ u l = v 1 ⋯ v m w 1 ⋯ w n , where u i ∈ {u, θ(u)} for all 1 ≤ i ≤ l, v j ∈ {v, θ(v)} for all 1 ≤ j ≤ m, w k ∈ {w, θ(w)} for all 1 ≤ k ≤ n, and θ is an antimorphic involution. More precisely, we show for which l, m, and n such an equation has only θ-periodic solutions, i.e., u,v,w ∈ {t,θ(t)}* for some word t, closing an open problem by Czeizler et al. (2009).