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The spectrum of integrable models is often encoded in terms of commutingfunctions of a spectral parameter that satisfy functional relations. We proposeto describe this commutative algebra in a covariant way by means of theextended Q-system that comprise Q-vectors in each of the fundamentalrepresentations of the (Langlands dual of) the underlying symmetry algebra.These Q-vectors turn out to parameterise a collection of complete flags whichare fused with one another in a particular way. We show that the fused flag isa finite-difference oper in a particular gauge, explicit identification dependson a choice of a Coxeter element.  The paper considers the case of simple Lie algebras with a simply-lacedDynkin diagram. For the $A_r$ series, the construction coincides with alreadyknown results in the literature. We apply the proposed formalism to the case ofthe $D_r$ series and the exceptional algebras $E_r$, $r=6,7,8$. In particular,we solve Hirota bilinear equations in terms of Q-functions and give theexplicit character solution of the extended Q-system in the $D_r$ case. We alsoshow how to build up the extended Q-system of $D_r$ type starting either fromvectors, by a procedure similar to the $A_r$ scenario which however constructsa fused flag of isotropic spaces, or from pure spinors, \via fused Fierzrelations.  Finally, for the case of rational, trigonometric, and elliptic spin chains,we propose an explicit ansatz for the analytic structure of Q-functions of theextended Q-system. We conjecture that the extended Q-system constrained in sucha way is always in bijection with the Bethe algebra of commuting transfermatrices of these models and moreover can be used to show that the Bethealgebra has a simple joint spectrum.

Extended systems of Baxter Q-functions and fused flags I: simply-laced case