A parametrization of a 1-dimensional curve is called regular if its velocity is always positive. For example, the following parametrization:
$$x(t)=t^3, y(t)=t^6$$
is not regular because its velocity is 0 in $t=0$.
But, this same curve can be re-parametrized as:
$$x(t)=t, y(t)=t^2$$
and this second parametrization is regular because its velocity anywhere is at least 1.
So, my question is: given a non-regular parametrization of a curve, is there an algorithm to tell whether the curve has a regular parametrization, and find it if it exists?