  Derivation-based differential calculi are of great importance in
noncommutative geometry, noncommutative gauge theory and integrable systems. In
this paper, we propose the connection and curvature from a class of deformed
derivation-based differential calculus. By means of this theory, we give out
the zero-curvature representation of the continuum-continuum,
discrete-continuum and discrete-discrete integrable systems in an unifying
manner.
