  We consider the class of higher derivative field equations whose wave
operator is a square of another self-adjoint operator of lower order. At the
free level, the models of this class are shown to admit a two-parameter series
of integrals of motion. The series includes the canonical energy. Every
conserved quantity is unbounded in this series. The interactions are included
into the equations of motion such that a selected representative in conserved
quantity series is preserved at the non-linear level. The interactions are not
necessarily Lagrangian, but they admit Hamiltonian form of dynamics. The theory
is stable if the integral of motion is bounded from below due to the
interaction. The motions are finite in the vicinity of the conserved quantity
minimum. The equations of motion for fluctuations have the derived form with no
resonance. The general constructions are exemplified by the models of the
Pais-Uhlenbeck oscillator with multiple frequency and Podolsky electrodynamics.
The example is also considered of stable non-abelian Yang-Mills theory with
higher derivatives.
