I canâ€™t find the exact paper I was thinking of but a starting point is Claudia Wulff and Frank Schilder. Numerical bifurcation of Hamiltonian relative periodic orbits.

SIAM J. Appl. Dyn. Syst., 8(3), pp. 931-966 (2009). (Iâ€™m not sure how accessible it is to people outside the field of Hamiltonian dynamics - certainly reading it now is a lot harder than I remember it being at the timeâ€¦)

Another example is Peeters, M., ViguiĂ©, R., SĂ©randour, G., Kerschen, G., & Golinval, J. C. (2009). Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques. Mechanical systems and signal processing, 23(1), 195-216. They construct an overdetermined system and use the Moore-Penrose pseudo-inverse in conjunction with a Gauss-Newton method (sec 3.2.2).

Related to the Peeters paper, the simplest example of adding extra variables I can think of is the computation of periodic orbits of the differential equation

\ddot x + x + x^3 = 0

This equation has a family of periodic orbits that live on a manifold. To find them the typical approach is to add a phase condition (to fix time invariance) and the pseudo-arclength equation (to allow path following), which results in f:\mathbb{R}^n\to\mathbb{R}^{n+1}. To solve this, either follow the paper and use Moore-Penrose with Newton or, alternatively, add the extra variable a\dot x to the differential equation, where a is to be solved for. This gives a new problem g:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1} which can be solved with standard techniques. The key to this is the fact that for a\neq0 the manifold does not exist and so the root finder will automatically choose a=0. The term a\dot x is known as an unfolding term; in this case it works because the manifold is a result of a conservation law in the underlying equations (in this case conservation of energy) and the unfolding term destroys the conservation law by adding dissipation. If your equations have some form of conservation law embedded within them, you might be able to add suitable unfolding terms but it isnâ€™t always obvious how to.