We consider freely decaying two-dimensional isotropic turbulence. It is usually assumed that, in such turbulence, the energy spectrum at small wavenumber, k, takes the form E(k → 0) ∼ Ik
3, where I is the two-dimensional version of Loitsyansky's integral. However, a second possibility is E(k → 0) ∼ Lk, where the pre-factor, L, is the two-dimensional analogue of Saffman's integral. We show that, as in three dimensions, L is an invariant and that E ∼ Lk spectra arise whenever the eddies possess a significant amount of linear impulse. The conservation of L is shown to be a direct consequence of the principle of conservation of linear momentum. We also show that isotropic turbulence dominated by a cloud of randomly located monopole vortices has a singular energy spectrum of the form E(k → 0) ∼ Jk
−1, where J, like L, is an invariant. However, while E ∼ Jk
−1 necessarily implies the existence of a sea of monopoles, the converse need not be true: a sea of monopoles whose spatial locations are not entirely random, but constrained in some way, need not give a E ∼ Jk
−1 spectra. The constraint imposed by the conservation of energy is particularly important, ruling out E ∼ Jk
−1 spectra for certain classes of initial conditions. Finally, we provide simple explicit examples of random vorticity fields with E ∼ Ik
3, E ∼ Lk and E ∼ Jk
−1 spectra.