  We introduce the notion of $N$-reflection equation which provides a large
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 to the rational and
trigonometric $r$-matrices. A central result is the construction of a Poisson
algebra associated to 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 $BC_L$-type .
