  We investigate spin-$\frac{1}{2}$ anisotropic model of a linear chain of
triangles with competing ferro- and antiferromagnetic interactions and
ferromagneic Heisenberg interactions between triangles. For a certain ratio
between interactions the one-magnon excitation band is dispersionless leading
to an existence of localized-magnon states which form macroscopically
degenerated ground states. The spectrum of excitations has a specific structure
depending on the value of the triangle-triangle interaction. Such specific
structure determines the low-temperature thermodynamics and, in particular, the
temperature dependence of a specific heat. In the limit of strong anisotropy of
interactions the spectrum has a multi-scale structure which consists of subsets
rank-ordered on small parameter. Each subset is responsible for the appearance
of the peak in the temperature dependence of the specific heat.
