  We consider the issue of correspondence between symmetries and conserved
quantities in the class of linear relativistic higher-derivative theories of
derived type. In this class of models the wave operator is a polynomial in
another formally self-adjoint operator, while each isometry of space-time gives
rise to the series of symmetries of action functional. If the wave operator is
given by n-th-order polynomial then this series includes n independent entries,
which can be explicitly constructed. The Noether theorem is then used to
construct an n-parameter set of second-rank conserved tensors. The canonical
energy-momentum tensor is included in the series, while the other entries
define independent integrals of motion. The Lagrange anchor concept is applied
to connect the general conserved tensor in the series with the original
space-time translation symmetry. This result is interpreted as existence of
multiple energy-momentum tensors in the class of derived systems. To study
stability we seek for bounded-conserved quantities that are connected with the
time translations. We observe that the derived theory is stable if its wave
operator is defined by a polynomial with simple and real roots. The general
constructions are illustrated by the examples of the Pais-Uhlenbeck oscillator,
higher-derivative scalar field, and extended Chern-Simons theory.
