  Let $(x(t),y(t))^\top$ be a solution of a Fuchsian system of order two with
three singular points. The vector space of functions of the form
$P(t)x(t)+Q(t)y(t)$, where $P,Q$ are real polynomials, has a natural filtration
of vector spaces, according to the asymptotic behaviour of the functions at
infinity. We describe a two-parameter class of Fuchsian systems, for which the
corresponding vector spaces obey the Chebyshev property (the maximal number of
isolated zeroes of each function is less than the dimension of the vector
space).
  Up to now, only a few particular systems were known to possess such a
nonoscillation property. It is remarkable that most of these systems are of the
type studied in the present paper. We apply our results in estimating the
number of limit cycles that appear after small polynomial perturbations of
several quadratic or cubic Hamiltonian systems in the plane.
