A dictionary of Confusable Phrases: More Than 10,000 Idioms and Collocations

Note: 
The expression is not equivalent in meaning to the phrase
keep one’s mind on something
—(also: have one’s mind on some-
thing) continue to pay attention to smth.; not be distracted from
smth.: She merely sat very still while he attended to the inter-
view he could no longer keep his mind on.
keep something in one’s head —be able to retain in memory
smth. such as facts, figures, etc.: I don’t know how the chair-
man keeps all these data in his head.
keep the ball rolling • keep the pot boiling
keep the ball rolling —(also: keep the wheels turning) keep an
activity, process, etc. going: The first step is to feed your body
healthy food so you’ll have the energy to keep the ball rolling.
keep the pot boiling —
1
. earn enough money to provide one’s
livelihood: Not only has doing portraits of people kept the
pot boiling, it has also led to my meeting some famous people.
2. = keep the ball rolling: We will keep the pot boiling on this
until David is released because this is a question of basic dem-
ocratic rights.
Note: 
Neither expression correlates in meaning with the phrase
keep the home fires burning
— keep things at home going as
usual (while smb. is away): These women kept the home fires
burning as they patiently waited until the wheels of the war
stopped turning and their husbands were home.
keep the seat warm • warm the bench
keep the seat warm —occupy a position temporarily for a
person who is not yet ready or qualified to hold it: They are
using Kim to just keep the seat warm till they find somebody
who can win them an election.
Note: 
The expression does not correlate in meaning with the
phrase keep one’s seat—(also: hold one’s seat) 
1
. remain seated:
Bartlet entered the room. Everyone started to stand, but the
Governor motioned for them to keep their seats. 2. keep from
falling: Though aided by the high-bowed “mountain-saddle”
I could scarcely keep my seat on horseback. 3. retain one’s posi-
tion, membership in parliament, etc.: Despite his bribery con-
viction, Nakamura kept his seat in two elections.
warm the bench —(Sport) sit among the substitute players
during a game: Aherne has warmed the bench much more
assiduously than he has passed the ball.
See also: bench warmer / chair warmer.