  We make a detailed analysis of the A-hypergeometric system (or GKZ system)
associated with a monomial curve and integral, hence resonant, exponents. We
characterize the Laurent polynomial solutions and show that these are the only
rational solutions. We also show that for any exponent, there are at most two
linearly independent Laurent solutions, and that the upper bound is reached if
and only if the curve is not arithmetically Cohen--Macaulay. We then construct,
for all integral parameters, a basis of local solutions in terms of the roots
of the generic univariate polynomial associated with A. We determine the
holonomic rank r for all integral exponents and show that it is constantly
equal to the degree d of X if and only if X is arithmetically Cohen-Macaulay.
Otherwise there is at least one exponent for which r = d + 1.
