  We investigate Lawruk elliptic boundary-value problems for homogeneous
differential equations in a two-sided refined Sobolev scale. These problems
contain additional unknown functions in the boundary conditions of arbitrary
orders. The scale consists of inner-product H\"ormander spaces whose orders of
regularity are given by any real number and a function which varies slowly at
infinity in the sense of Karamata. We establish theorems on the Fredholm
property of the problems in the refined Sobolev scale and on local regularity
and local a priori estimate (up to the boundary of the domain) of their
generalized solutions. We find sufficient conditions under which components of
these solutions are $l\geq0$ times continuously differentiable functions.
