  We consider analytically and numerically an anisotropic spin-$\frac{1}{2}$
delta-chain (sawtooth chain) with nearest-neighbor ferromagnetic and
next-nearest-neighbor antiferromagnetic interactions. For certain values of the
interactions a lowest one-particle band becomes flat and there is a class of
localized-magnon eigenstates which form a ground state with a macroscopic
degeneracy. In this case the model depends on a single parameter which can be
chosen as the anisotropy of the exchange interactions. When this parameter
changes from zero to infinity the model interpolates between the
one-dimensional isotropic ferromagnet and the frustrated Ising model on the
delta-chain. It is shown that the low-temperature thermodynamic properties in
these limiting cases are governed by the specific structure of the excitation
spectrum. In particular, the specific heat has one or infinite number of
low-temperature maxima for the small or the large anisotropy parameter,
correspondingly. Numerical calculations of finite chains demonstrate that this
behavior is generic for definite values of the anisotropy parameter.
