  In a refined Sobolev scale, we investigate an elliptic boundary-value problem
with additional unknown functions in boundary conditions for which the maximum
of orders of boundary operators is grater than or equal to the order of the
elliptic equation. This scale consists of inner product H\"ormander spaces
whose order of regularity is given by a real number and a function varying
slowly at infinity in the sense of Karamata. We prove a theorem on the Fredholm
property of a bounded operator corresponding to this problem in the refined
Sobolev scale. For the generalized solutions to the problem, we establish a
local a priory estimate and prove a theorem about their regularity in
H\"ormander spaces. We find sufficient conditions under which given generalized
derivatives of the solutions are continuous.
