We describe relationships between integrable systems with N degrees of
freedom arising from the AGT conjecture. Namely, we prove the equivalence
(spectral duality) between the N-cite Heisenberg spin chain and a reduced gl(N)
Gaudin model both at classical and quantum level. The former one appears on the
gauge theory side of the AGT relation in the Nekrasov-Shatashvili (and further
the Seiberg-Witten) limit while the latter one is natural on the CFT side. At
the classical level, the duality transformation relates the Seiberg-Witten
differentials and spectral curves via a bispectral involution. The quantum
duality extends this to the equivalence of the corresponding Baxter-Schrodinger
equations (quantum spectral curves). This equivalence generalizes both the
spectral self-duality between the 2x2 and NxN representations of the Toda chain
and the famous AHH duality