# Generalised Lyndon-Schützenberger Equations

We fully characterise the solutions of the generalised Lyndon-Schützenberger word equations $$u_1 \cdots u_\ell = v_1 cdots v_m w_1 \cdots w_n$$, where $$u_i \in \{u, \theta(u)\}$$ for all $$1 \leq i \leq \ell$$, $$v_j \in \{v, \theta(v)\}$$ for all $$1 \leq j \leq m$$, $$w_k \in \{w, \theta(w)\}$$ for all $$1 \leq k ?\leq n$$, and $$\theta$$ is an antimorphic involution. More precisely, we show for which $$\ell$$, $$m$$, and $$n$$ such an equation has only $$\theta$$-periodic solutions, i.e., $$u$$, $$v$$, and $$w$$ are in $$\{t, \theta(t)\}^\ast$$ for some word $$t$$, closing an open problem by Czeizler et al. (2011).

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