  The Nested Bethe Ansatz is generalized to open and independent boundary
conditions depending on two continuous and two discrete free parameters. This
is used to find the exact eigenvectors and eigenvalues of the $A_{n-1}$ vertex
models and $SU(n)$ spin chains with such boundary conditions. The solution is
found for all diagonal families of solutions to the reflection equations in all
possible combinations. The Bethe ansatz equations are used to find de first
order finite size correction.
