  Since the last century, considerable efforts have been devoted to the study
of valley-degenerate narrow gap semiconductors, such as the Pb$_{1-x}$Sn$_x$Se
alloy. This material possesses band minima at the $L$-points of their Brillouin
zone, yielding a valley degeneracy of four. However, in (111)-oriented films,
it is still not fully understood how differences between the longitudinal
valley, oriented along the growth axis, and the oblique valleys, oriented at an
angle with respect to that axis, appear in magneto-optical infrared
spectroscopy. In this work, we report a magnetooptical study on this family of
alloys, focusing on an anomaly in the interband transition of the peak
intensity ratio of longitudinal and oblique valleys under a magnetic field
applied along the [111] direction. Based on the Mitchell-Wallis model, we
provide a theoretical fit for the experimental transmission data, which
quantitatively explains the spectral shape of the data at magnetic fields as
high as 35T. In particular, we attribute this anomalous peak intensity
variation to the carrier density difference between the two types of valleys as
well as the field-dependent thin-film interference. Our analysis also allows
for the extraction of the real and imaginary parts of the dielectric function.
