This chapter does not prohibit the independent nomination of candidates under Part 2 (commencing with Section 8300), subject to the following limitations:
(a) A candidate whose name has been on the ballot as a candidate of a party at the direct primary and who has been defeated for that party nomination is ineligible for nomination as an independent candidate. He is also ineligible as a candidate named by a party central committee to fill a vacancy on the ballot for a general election.
(b) No person may file nomination papers for a party nomination and an independent nomination for the same office, or for more than one office at the same election.