Common framework and quadratic Bethe equations for rational Gaudin magnets in arbitrarily oriented magnetic fields

In this work we demonstrate a simple way to implement the quantum inverse scattering
method to find eigenstates of spin-1/2 XXX Gaudin magnets in an arbitrarily oriented
magnetic field. The procedure differs vastly from the most natural approach which would
be to simply orient the spin quantisation axis in the same direction as the magnetic field
through an appropriate rotation.
Instead, we define a modified realisation of the rational Gaudin algebra and use the
quantum inverse scattering method which allows us, within a slightly modified imple-
mentation, to build an algebraic Bethe ansatz using the same unrotated reference state
(pseudovacuum) for any external field. This common framework allows us to easily write
determinant expressions for certain scalar products which would be highly non-trivial
in the rotated system approach.