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The main results of this paper are as follows: (1) Suppose an s stage RungeKutta method is consistent, irreducible, non-confluent and symplectic. Then this method is of order at least 2p +l(1 < p < s - 1) provided that the simplifying conditions C(p) (or D(p) with non-zero weights) and B(2p + l) hold, where l=0, 1, 2. (2) Suppose an s stage Runge-Kutta method is consistent, irreducible and non-confluent, and satisfies the simplifying conditions C(p) and D(p) with 0 ＜ p
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