  We prove the existence of a family of non-trivial solutions of the Liouville
equation in dimensions two and four with infinite volume. These solutions are
perturbations of a finite-volume solution of the same equation in one dimension
less. In particular, they are periodic in one variable and decay linearly to
$-\infty$ in the other variables. In dimension two, we also prove that the
periods are arbitrarily close to $\pi k, k \in \mathbb{N}$ (from the positive
side). The main tool we employ is bifurcation theory in weighted H\"older
spaces.
