$\tau^2 =\varphi_b\varphi_\beta $.

\[nerozlisuj\] Two infinite words $\uu$ and $E(\uu)$ over the alphabet $\{0,1\}$ coincide up to a permutation of the letters $0$ and $1$. If a word $\uu$ is a fixed point of a morphism $\varphi_w$, then $E(\uu)$ is a fixed point of the morphism $E\circ \varphi_w\circ E = \varphi_v$ for some $v$. By , the word $v$ is obtained from $w$ by exchange of letters $a\leftrightarrow \alpha$ and $b\leftrightarrow\beta$. Therefore we introduce the following morphism $F: \{a,b,\alpha,\beta\}^* \mapsto \{a,b,\alpha,\beta\}^* $ by $$\label{eq:definition_of_F}
F(a)=\alpha, \quad F(\alpha)=a,\quad F(b)=\beta, \quad F(\beta)=b.$$ This notation enables us to formulate two useful facts on composition of $E$ with morphisms from $\mathcal{M}$. Namely, $$\label{skladani} E\circ \varphi_w\circ E = \varphi_{F(w)} \qquad \text{ and} \qquad (\varphi_w\circ E)^2 = \varphi_{wF(w)}\,.$$

Later on we will need the following statement on the morphism $F$. First we recall two classical results on word equations:

\[lem:Lyndon1\] Let $y \in \A^*$ and $x,z \in \A^+$. Then $xy = yz$ if and only if there are $u, v \in \A^*$ and $\ell \in \N$ such that $x = uv, z = vu$ and $y = (uv)^\ell u$.

\[lem:Lyndon2\] Let $x, y \in \A^+$. The following three conditions are equivalent:

(i) $xy  = yx$;

(ii) 