  We consider a class of self-adjoint extensions using the boundary triple
technique. Assuming that the associated Weyl function has the special form
$M(z)=\big(m(z)\Id-T\big) n(z)^{-1}$ with a bounded self-adjoint operator $T$
and scalar functions $m,n$ we show that there exists a class of boundary
conditions such that the spectral problem for the associated self-adjoint
extensions in gaps of a certain reference operator admits a unitary reduction
to the spectral problem for $T$. As a motivating example we consider
differential operators on equilateral metric graphs, and we describe a class of
boundary conditions that admit a unitary reduction to generalized discrete
laplacians.
