The moduli problem for (algebraic completely) integrable systems is introduced. This problem consists
in constructing a moduli space of affine algebraic varieties and explicitly describing a map which associates
to a generic affine variety, which appears as a level set of the first integrals of the system (or, equivalently,
a generic affine variety which is preserved by the flows of the integrable vector fields), a point in this moduli
space. As an illustration, the example of an integrable geodesic flow on SO(4) is worked out. In this case,
the generic invariant variety is an affine part of the Jacobian of a Riemann surface of genus 2. The
construction relies heavily on the fact that these affine parts have the additional property of being 4:1
unramified covers of Abelian surfaces of type (1,4).