Consider Context-Free Grammars with rules of the form $S\to\epsilon$or $A\to aB | Ba|a$,meaning that any nonterminal other than $S$ can be replaced by a terminal, or by a pair of letters exactly one of which is nonterminal (and the other one terminal). What class of languages do such grammars generate? Certainly the regular ones, but also $a^n b^n$, via the rules $S\to\epsilon|aB$, $B\to b|Ab$, $A\to a|aB$. Short of a full characterization, I conjecture that these grammars do not generate the full CFL language family, and would be curious to see a proof.
CFGs in a certain normal form
fl.formal languages;automata theory
A linear grammar is a context-free grammar that has at most one nonterminal in the right hand side of each of its productions.A linear language is a language generated by some linear grammar.See also:  https://en.wikipedia.org/wiki/Linear_grammarIt is known that the regular languages form a proper subfamily of the linear languages, and that the linear languages form a proper subfamily of the context-free languages.The language $\{a^kb^kc^nd^n|k,n\ge1\}$ is known to be context-free but not linear.The text-book Introduction to Automata Theory, Languages, and Computation by John E. Hopcroft and Jeffrey D. Ullman (Addison-Wesley, 1979) states a pumping lemma for linear languages:Let $L$ be a linear language.  Then there exists an integer $n$ such that for any word $p\in L$ with $|p|\ge n$, there exists a factorization $p=uvwxy$ with the following properties: $uv^iwx^iy \in L$ for all integers $i\in\mathbb{N}$$|vx|>0$$|uvxy| \le n$.