  We analyze the spectral properties of a phenomenological model for a weakly
doped two-dimensional antiferromagnet, in which the carriers move within one of
the two sublattices where they were introduced. Such a constraint results in
the free carrier spectra with the maxima at ${\bf k}=(\pm \pi /2 ,\pm \pi /2)$
observed in some cuprates. We consider the spectral properties of the model by
taking into account fluctuations of the spins in the antiferromagnetic
background. We show that such fluctuations lead to a non-pole-like structure of
the single-hole Green's function and these fluctuations can be responsible for
some anomalous "strange metal" properties of underdoped cuprates in the
nonsuperconducting regime.
