Twenty-second Amendment to the United States Constitution
Some argue that the 22nd Amendment and 12th Amendment bar any two-term president from later serving as vice president as well as from succeeding to the presidency from any point in the presidential line of succession. Others contend that the original intent of the 12th Amendment concerns qualification for "service" (age, residence, and citizenship), while the 22nd Amendment, concerns qualifications for "election", and thus (strictly applying the text) a former two-term president is still eligible to "serve" as vice president (neither amendment restricts the number of times an individual can be elected to the vice presidency), and then succeed to the presidency to serve out the balance of the term (though prohibited from running for election to an additional term).
Resign-to-run laws
Tex. Elec. Code § 145.001(e) permits a person to run for office and simultaneously be a candidate for President or Vice President of the United States. This statute permitted Lyndon B. Johnson to run for Vice President in 1960 and, at the same time, seek re-election as United States Senator from Texas. Lloyd Bentsen took advantage of the same provision in 1988 when he was the Vice Presidential running mate of Michael Dukakis.