Can a Statutoty auditor maintain books of accounts

No, auditor cannot maintain books of account

No a statutory auditor shall not provide book keeping services as per section 144

As Per Section 144 Of Companies Act 2013,
An Auditor Can Never Maintain Books Of Accounts Of The SAME Company.
However Law Never Restricts An Auditor To Maintain Books Of Accounts For Any Other Company.

As per sec 144, If Board of directors of the company approves the auditors of the company to provide services other than audit like; Court case, legal petition etc. are allowed but certain services related to financials, Books, Consultation, Management etc. are not allowed.
Logic behind the same is - Auditor is to check the legal aspects , financials, Books etc. and to provide an opinion whether books are properly maintained. And if, he will be the maker as well as the checker, then Audit will be of no use.