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Combined electric and heat systems can significantly improve fuel efficiency and reduce carbon emission by making use of complementary properties of electricity and heat. However, many current papers on the energy market mechanism do not consider the heating price, while some do not address the challenges of heat time delays on prices, which may impede the benefits of system operators as well as users. Therefore, in this paper, we extend the electric local marginal price (LMP) to the integrated LMP (ILMP) for electricity and heat in combined electric and heat systems considering heat time delays and network constraints. The results of case studies demonstrate the proposed mechanism can provide fair electricity and heat prices which can reflect generation cost, power loss, network congestion, and time delays.