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Any infinite uniformly recurrent word ${\bf u}$ can be written asconcatenation of a finite number of return words to a chosen prefix $w$ of${\bf u}$. Ordering of the return words to $w$ in this concatenation is codedby derivated word $d_{\bf u}(w)$. In 1998, Durand proved that a fixed point${\bf u}$ of a primitive morphism has only finitely many derivated words$d_{\bf u}(w)$ and each derivated word $d_{\bf u}(w)$ is fixed by a primitivemorphism as well. In our article we focus on Sturmian words fixed by aprimitive morphism. We provide an algorithm which to a given Sturmian morphism$\psi$ lists the morphisms fixing the derivated words of the Sturmian word${\bf u} = \psi({\bf u})$. We provide a sharp upper bound on length of thelist.

Fixed points of Sturmian morphisms and their derivated words