  We study a class of evolutionary partial differential systems with two
components related to second order (in time) non-evolutionary equations of odd
order in spatial variable. We develop the formal diagonalisation method in
symbolic representation, which enables us to derive an explicit set of
necessary conditions of existence of higher symmetries. Using these conditions
we globally classify all such homogeneous integrable systems, i.e. systems
which possess a hierarchy of infinitely many higher symmetries.
