This paper constructs a novel multi-objective optimal energy flow of an integrated electricity and gas system to fully exploit complementary benefits of the system. To ensure a reliable convergence, an incremental piecewise linearization is employed to linearize the original nonlinear problem into a mixed integer linear programming. Actually, the proposed issue represents a highly constrained optimization with numerous variables and multiple conflicting objectives. To effectively solve the problem, a novel analytical Pareto optimizer called homogenized adjacent points method (HAPM) is first proposed, which aims to obtain a Pareto solution set with three basic strategies, including axes homogenization, adjacent points calculation and adjacent points filtration. Compared to the existing methods, the major superiority of HAPM is that the algorithm can obtain a full distribution of Pareto solution set with the boundary of Pareto front and non-dominated quality of the solutions. Finally, a best compromise solution is identified from the Pareto solution set using an entropy weight based ideal point method. Simulation results indicate an environmentally friendly and reliable integrated electricity and gas system is realized with a slight cost sacrifice, and validate the robustness of HAPM to obtain a non -dominated, uniform and widespread Pareto solution set compared with existing methods.