  We present the extension of our \cocal~- Compact Object CALculator - code to
compute general-relativistic initial data for binary compact-star systems. In
particular, we construct quasiequilibrium initial data for equal-mass binaries
with spins that are either aligned or antialigned with the orbital angular
momentum. The Isenberg-Wilson-Mathews formalism is adopted and the constraint
equations are solved using the representation formula with a suitable choice of
a Green's function. We validate the new code with solutions for equal-mass
binaries and explore its capabilities for a wide range of compactnesses, from a
white dwarf binary with compactness $\sim 10^{-4}$, up to a highly relativistic
neutron-star binary with compactness $\sim 0.22$. We also present a comparison
with corotating and irrotational quasiequilibrium sequences from the spectral
code \lorene [Taniguchi and Gourgoulhon, Phys. Rev. D {\bf 66}, 104019 (2002)]
and with different compactness, showing that the results from the two codes
agree to a precision of the order of $0.05\%$. Finally, we present equilibria
for spinning configurations with a nuclear-physics equation of state in a
piecewise polytropic representation.
