A new degree function, defined for flows which have a continuously differentiable first integral, counts, algebraically, the number of orbits of fixed period τ in a set Ω. The degree takes account of the number of such orbits and of the order of their isotropy group. The context in which it is defined, namely when a dynamical system has a first integral, is one where the Fuller index is always trivial. In passing we give a rudimentary account of generic bifurcation theory for orbits of fixed period of dynamical systems which have a first integral. The paper is in two parts. The first gives a reasonably self-contained account of the principles involved in the definition of the degree function and of the consequent degree theory, which should be accessible to a wide audience including those with an interest in applications. Part 2 is a highly technical detailed account of the proofs that all the claims made in Part 1 are valid.
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