We study symmetries of the Bethe equations for the Gaudin model. These symmetries arise naturally under the name Hecke operator
in the framework of the geometric Langlands correspondence and under the name Schlesinger transformation in the theory of
isomondromy deformations, in particular, in the theory of Painlevé transcendents.
Keywords Gaudin model - Bethe ansatz - isomondromy transformation
Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 159, No. 2, pp. 252–265, May, 2009.