The oscillatory growth of a dilute binary alloy has recently been described by a nonlinear oscillator equation that applies to small temperature gradients and large growth velocities in the setup of directional solidification. Based on a one-dimensional stability analysis of stationary solutions of this equation, we explore in the present paper the complete region where the solidification front propagates in an oscillatory way. The boundary of this region is calculated exactly, and the nature of the oscillations is evaluated numerically in several segments of the region.