My recollection from studies of a bygone past is that the "rule" about not splitting infinitives derives from the import of Latin grammar into English (when was that? I dunno ... I wasn't there, although I feel like I could have been). The rule is occasioned because in Latin the infinitive form is always a single word. See below for examples. Note there are multiple forms of infinitives for different tenses.
"To sing" =
present infinitive active canere to sing
present infinitive passive cani to be sung
perfect infinitive active cecinisse to have sung
perfect infinitive passive cantum -am -um -os -as -a esse to have been sung
future infinitive active canturum -am -um -os -as -a esse to be about to sing
future infinitive passive cantum iri to be about to be sung