In this article we use the Maple package Jets to compute symmetry groups of the Novikov equation and the Geng-Xue system. The group generators correspond to transformations that take solutions of the equations to other solutions. By applying one of the transformations for the Novikov equation to the known one-peakon solution, we find a new kind of unbounded solutions with peakon creation, i.e., these functions are smooth solutions for some time interval, then after a certain finite time, a peak is created. We show that the functions are still weak solutions to the Novikov equation for those times where the peak lives. We also find similar unbounded solutions with peakon creation in the related Camassa-Holm equation, by making an ansatz inspired by the Novikov solutions. Finally, we see that the same ansatz for the Degasperis-Procesi equation yields unbounded solutions where a peakon is present for all times.