  The paper studies the problem of making Getz's bicycle model traverse a
strictly convex Jordan curve with bounded roll angle and bounded speed. The
approach to solving this problem is based on the virtual holonomic constraint
(VHC) method. Specifically, a VHC is enforced making the roll angle of the
bicycle become a function of the bicycle's position along the curve. It is
shown that the VHC can be automatically generated as a periodic solution of a
scalar periodic differential equation, which we call virtual constraint
generator. Finally, it is shown that if the curve is sufficiently long as
compared to the height of the bicycle's centre of mass and its wheel base, then
the enforcement of a suitable VHC makes the bicycle traverse the curve with a
steady-state speed profile which is periodic and independent of initial
conditions. An outcome of this work is a proof that the constrained dynamics of
a Lagrangian control system subject to a VHC are generally not Lagrangian.
