  The noncommutative space $\mathbb{R}^3_\lambda$, a deformation of
$\mathbb{R}^3$, supports a $3$-parameter family of gauge theory models with
gauge-invariant harmonic term, stable vacuum and which are perturbatively
finite to all orders. Properties of this family are discussed. The partition
function factorizes as an infinite product of reduced partition functions, each
one corresponding to the reduced gauge theory on one of the fuzzy spheres
entering the decomposition of $\mathbb{R}^3_\lambda$. For a particular
sub-family of gauge theories, each reduced partition function is exactly
expressible as a ratio of determinants. A relation with integrable 2-D Toda
lattice hierarchy is indicated.
