We start with elementary algebraic theory of factorization of linear ordinary differential equations developed in the period 1880-1930. After exposing these classical results we sketch more sophisticated algorithmic approaches developed in the last 2
0 years. The main part of this paper is devoted to modern generalizations of the notion of factorization to the case of systems of linear partial differential equations and their relation with explicit solvability of nonlinear partial differential equations based on some constructions from the theory of abelian categories.
We classify all integrable 3-dimensional scalar discrete quasilinear equations Q=0 on an elementary cubic cell of the 3-dimensional lattice. An equation Q=0 is called integrable if it may be consistently imposed on all 3-dimensional elementary faces
of the 4-dimensional lattice. Under the natural requirement of invariance of the equation under the action of the complete group of symmetries of the cube we prove that the only nontrivial (non-linearizable) integrable equation from this class is the well-known dBKP-system. (Version 2: A small correction in Table 1 (p.7) for n=2 has been made.) (Version 3: A few small corrections: one more reference added, the main statement stated more explicitly.)
We study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by respective discrete nets (circular nets and planar quadrilateral nets) with infinitesimal quads. It is shown that choosing the points of discrete
nets on the smooth surface one can obtain second-order approximation globally. Also a simple geometric construction for approximate determination of principal directions of smooth surfaces is given.
