  This paper is devoted to the analysis of linear second order discrete-time
descriptor systems (or singular difference equations (SiDEs) with control).
Following the algebraic approach proposed by Kunkel and Mehrmann for pencils of
matrix valued functions, first we present a theoretical framework based on a
procedure of reduction to analyze solvability of initial value problems for
SiDEs, which is followed by the analysis of descriptor systems. We also
describe methods to analyze structural properties related to the solvability
analysis of these systems. Namely, two numerical algorithms for reduction to
the so-called strangenessfree forms are presented. Two associated index notions
are also introduced and discussed. This work extends and complements some
recent results for high order continuous-time descriptor systems and first
order discrete-time descriptor systems.
