"keep verb-ing" and "keep on verb-ing" are both correct. In most cases there is not a great difference in meaning, but the version with "on" has slightly more emphasis and feels more colloquial.
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Anonymousthe usage of 'on'.Your sentences are all correct, but 'keep' can be used without 'on', so I would omit it because there is another 'on' quite close in the same sentence.