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The Spectral Problem is to describe possible spectra $\sigma (A_j)$ for anirreducible $n$-tuple of Hermitian operators s.t. $A_1+...+A_n$ is a scalaroperator. In case when $m_j= | \sigma (A_j)|$ are finite and a rooted tree${\rm T}_{m_1,..., m_n}$ with $n$ branches of lengths $m_1, ..., m_n$ is aDynkin graph the explicit answer to the Spectral Problem was given recently byS. A. Kruglyak, S. V. Popovych, and Yu. S. Samo\v\i{}lenko. In present work thesolution of the Spectral Problem for all star-shaped simply laced extendedDynkin graphs, i.e. when $(m_1, ..., m_n) \in \{(2,2,2,2), (3,3,3), (4,4,2)$,$(6,3,2)\}$ is presented.

The Spectral Problem and Algebras Associated with Extended Dynkin Graphs