A systematic way of obtaining integrable reductions of a classical system investigated by Darboux (1887-96) in connection with conjugate coordinate systems is presented. It includes, in particular, the Lame system, its generalization to pseudo-Riemannian spaces of constant curvature, an integrable 2+1-dimensional sine-Gordon equation and a hyperbolic equation of Klein-Gordon type. The integrability of a classical generalized Weingarten system set down by Bianchi (1957) is proven by means of a suitable superposition of two constraints. It is shown that these reductions are preserved under a Darboux-Levi-type transformation. A connection to the Moutard transformation is recorded.