  This paper deals with differential pencils possessing a term depending on the
unknown function with a fixed argument. We deduce the so called main equation
together with its fine structure for the spectral problem. Then, according to
the boundary conditions and the position of argument, we describe two cases:
degenerate and non-degenerate. For these two cases, the uniqueness of inverse
spectral problem are studied and a constructive procedure for reconstructing
the potentials along with necessary and sufficient conditions of its
solvability are obtained.
