Geodesics on the space of normalized univalent functions defined on the unit disc can be lifted to geodesics on the group of orientation preserving diffeomorphisms of the unit circle, under the condition of respecting some non-holonomic constraints. Thus, we aim to find non-holonomic geodesic equations on the infinite dimensional Lie groups having in mind the model example of the Virasoro-Bott group and the group of sense preserving diffeomorphisms of the unit circle. The non-holonomic geodesic equations are a generalization of inviscid Burgers', Camassa-Holm, Hunter-Saxton, KdV equations, and other known non-linear PDE related to the fluid mechanics.