Context: "They struggled to keep (to) (the) rule, not letting some students go out in recess or using corporal punishment."
Is "keep rule" a common set phrase in AmE? I've found the phrase in an autobiography.
As a mass noun, Lexico gives
rule
Control of or dominion over an area or people.
There is no stipulation of a time limit for US political rule and economic control over the country's resources.
It isn't a very common way of saying 'keep control'.
"Keep to rule" is like "working to rule", that is, strictly by the book.
"Keep the rule" refers to practising a particular rule.