The purpose of this paper is to study the connections existing between Serre's reduction of linear functional systems -- which aims at finding an equivalent system defined by fewer equations and fewer unknowns -- and the decomposition problem -- which aims at finding an equivalent system having a diagonal block structure -- in which one of the diagonal blocks is assumed to be the identity matrix. In order to do that, we further develop results on Serre's reduction problem and on the decomposition problem previously obtained. Finally, we show how these techniques can be used to analyze the decomposability problem of standard linear systems of partial differential equations studied in hydrodynamics such as Stokes equations, Oseen equations and the movement of an incompressible fluid rotating with a small velocity around the vertical axis.