  We address a generalized Richardson model (Russian doll BCS model), which is
characterized by the breaking of time-reversal symmetry. This model is known to
be exactly solvable and integrable. We point out that the Russian doll BCS
model, on the level of Hamiltonian, is also particle-hole symmetric. This
implies that the same state can be expressed both in the particle and hole
representations with two different sets of Bethe roots. We then derive exact
relations between Bethe roots in the two representations, which can hardly be
obtained staying on the level of Bethe equations. In a quasi-classical limit,
similar identities for usual Richardson model, known from literature, are
recovered from our results. We also show that these relations for Richardson
roots take a remarkably simple form at half-filling and for a symmetric with
respect to the middle of the interaction band distribution of one-body energy
levels, since, in this special case, the rapidities in the particle and hole
representations up to the translation satisfy the same system of equations.
