The operator i∂3+i∂p+ip∂+q with 1-periodic coefficients p, q ∈ L1 loc(R) is considered on the real line. The following results are obtained: 1) the spectrum of this operator is absolutely continuous, covers the entire real line, and has multiplicity one or three; 2) the spectrum of multiplicity three is bounded and expressed in terms of real zeros of a certain entire function; 3) the Lyapunov function, analytic on a 3-sheeted Riemann surface, is constructed and investigated.