As seen in this question, the class of languages that can be generated by a context-free grammar having only one non-terminal symbol (i.e. the start symbol) is a proper subclass of the class of context-free languages (in particular, it doesn't contain and is not contained in the class of regular languages).
I'd like to know if there is a commonly used name for this subclass, and more importantly, if it is decidable whether a language $L$ generated by some context-free grammar $G$ belongs to this class or not (maybe under some assumptions on $G$).
Any reference to related topics would be very appreciated. Thanks.