This paper studies systems of
linear homogeneous p.d.e. in two independent variables with constant coefficients.
For such systems powerful algebraic tools are available to obtain results which may
indicate patterns for more general systems. Linear isomorphism is defined,
and necessary and sufficient conditions for linear isomorphism between two
systems are found. This result is obtained from the infinite prolongation of
the systems, and two systems are isomorphic if and only if their infinite
prolongations are isomorphic. One unexpected result is the important role played by
lower-order coefficients which do not appear in such classical motions as
ellipticity, hyperbolicity or characteristics. The classification problem for these
p.d.e. is reduced to a problem in linear algebra involving a finite number of
relations.
