Automata and Formal Languages: Are there also non-regular grammars for regular languages?

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In Automata and Formal Languages, are there also grammars that are not right-/left-linear but still produce a regular language?

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The following non-regular grammar generates the regular language $01^*01^*01^*$; however, it could be transformed into a regular grammar.

$S \to 0A0A0A$
$A \to 1A|\varepsilon$