The Gaussian model for the factors leads to inability to separate the underlying causes from each other. This is remedied by considering models where the factors are not restricted to be Gaussian but have a more general model. These models can be called independent factor analysis (IFA) models or independent component analysis (ICA) models depending whether they are seen as extensions of factor analysis or principal component analysis models (for textbook accounts see, e.g., [76,34]).
The independence refers to the fact that the non-Gaussian model of the factors enables factor analysis to find the underlying statistically independent factors which have generated the observations linearly if they exist. In the ICA community, the factors have been traditionally called sources because they are not just some arbitrary combinations of the underlying causes, but ideally at least, the original independent causes. The practical techniques for estimating the sources are known also as blind source or signal separation.
Other properties apart from fixing the rotation of the factors are largely the same for the linear non-Gaussian factor analysis as for the linear Gaussian factor analysis. A linear multi-layer model, for instance, can still be brought down into one linear layer.