  By means of transformations to nonlinear Klein-Gordon equations, we show that
a generalized short pulse equation is integrable in two (and, most probably,
only two) distinct cases of its coefficients. The first case is the original
short pulse equation (SPE). The second case, which we call the single-cycle
pulse equation (SCPE), is a previously overlooked scalar reduction of a known
integrable system of coupled SPEs. We get the Lax pair and bi-Hamiltonian
structure for the SCPE and show that the smooth envelope soliton of the SCPE
can be as short as only one cycle of its carrier frequency.
