  We analyse an algorithm of transition between Cauchy problems for
second-order wave equations and first-order symmetric hyperbolic systems in
case the coefficients as well as the data are non-smooth, even allowing for
regularity below the standard conditions guaranteeing well-posedness. The
typical operations involved in rewriting equations into systems are then
neither defined classically nor consistently extendible to the distribution
theoretic setting. However, employing the nonlinear theory of generalized
functions in the sense of Colombeau we arrive at clear statements about the
transfer of questions concerning solvability and uniqueness from wave equations
to symmetric hyperbolic systems and vice versa. Finally, we illustrate how this
transfer method allows to draw new conclusions on unique solvability of the
Cauchy problem for wave equations with non-smooth coefficients.
