The behaviour of a space-modulated, so-called " argumental " oscillator is studied, which is represented by a model having an even-parity space-modulating function. Analytic expressions of a stability criterion and of discrete energy levels are given. Using an integrating factor and a Van der Pol representation in the (amplitude, phase) space, an approximate implicit closed-form of the solution is given. The probability to enter a stable-oscillation regime from given initial conditions is calculated in symbolic form. These results allow an analytic approach to stability and bifurcations of the system. They also allow an assessment of the risk of occurrence of sustained large-amplitude oscillations, when the phenomenon is to be avoided, and an assessment of the conditions to apply to obtain oscillations whenever the phenomenon is desired.