  We consider the simplest nontrivial supersymmetric quantum mechanical system
involving higher derivatives. We unravel the existence of additional bosonic
and fermionic integrals of motion forming a nontrivial algebra. This allows one
to obtain the exact solution both in the classical and quantum cases. The
supercharges $Q, \bar Q$ are not anymore Hermitially conjugate to each other,
which allows for the presence of negative energies in the spectrum. We show
that the spectrum of the Hamiltonian is unbounded from below. It is discrete
and infinitely degenerate in the free oscillator-like case and becomes
continuous running from $-\infty$ to $\infty$ when interactions are added.
Notwithstanding the absence of the ground state, there is no collapse, which
suggests that a unitary evolution operator may be defined.
