  In this paper we generalise a useful result due to J. Mierczynski which
states that for a strictly cooperative system on the positive orthant, with
increasing first integral, all bounded orbits are convergent. Moreover any
equilibrium attracts its entire level set, and there can be no more than one
equilibrium on any level set. Here, more general state spaces and more general
orderings are considered. Let Y subset K subset R^n be any two proper cones.
Given a local semiflow phi on Y which is strongly monotone with respect to K,
and which preserves a K-increasing first integral, we show that every bounded
orbit converges. Again, each equilibrium attracts its entire level set, and
there can be no more than one equilibrium on any level set. An application from
chemical dynamics is provided.
