  We consider the operator $H={d^4dt^4}+{ddt}p{ddt}+q$ with 1-periodic
coefficients on the real line. The spectrum of $H$ is absolutely continuous and
consists of intervals separated by gaps. We describe the spectrum of this
operator in terms of the Lyapunov function, which is analytic on a two-sheeted
Riemann surface. On each sheet the Lyapunov function has the standard
properties of the Lyapunov function for the scalar case.
  We describe the spectrum of $H$ in terms of periodic, antiperiodic
eigenvalues, and so-called resonances.
  We prove that 1) the spectrum of $H$ at high energy has multiplicity two, 2)
the asymptotics of the periodic, antiperiodic eigenvalues and of the resonances
are determined at high energy, 3) for some specific $p$ the spectrum of $H$ has
an infinite number of gaps, 4) the spectrum of $H$ has small spectral band
(near the beginner of the spectrum) with multiplicity 4 and its asymptotics are
determined as $p\to 0, q=0$.
