Classical N-reflection equation and Gaudin models

We introduce the notion of N-reflection equation which provides a generalization of the usual classical reflection equation describing integrable boundary conditions. The latter is recovered as a special example of the N=2 case. The basic theory is established and illustrated with several examples of solutions of the N-reflection equation associated with the rational and trigonometric r-matrices. A central result is the construction of a Poisson algebra associated with a non-skew-symmetric r-matrix whose form is specified by a solution of the N-reflection equation. Generating functions of quantities in involution can be identified within this Poisson algebra. As an application, we construct new classical Gaudin-type Hamiltonians, particular cases of which are Gaudin Hamiltonians of BCL-type.