  We study a new generalized version of the square-lattice frustrated XY model
where unequal ferromagnetic and antiferromagnetic couplings are arranged in a
zig-zag pattern. The ratio between the couplings $\rho$ can be used to tune the
system, continuously, from the isotropic square-lattice to the
triangular-lattice frustrated XY model. The model can be physically realized as
a Josephson-junction array with two different couplings, in a magnetic field
corresponding to half-flux quanta per plaquette. Mean-field approximation,
Ginzburg-Landau expansion and finite-size scaling of Monte Carlo simulations
are used to study the phase diagram and critical behavior. Depending on the
value of $\rho$, two separate transitions or a transition line in the
universality class of the XY-Ising model, with combined $Z_2$ and U(1)
symmetries, takes place. In particular, the phase transitions of the standard
square-lattice and triangular-lattice frustrated XY models correspond to two
different cuts through the same transition line. Estimates of the chiral
($Z_2$) critical exponents on this transition line deviate significantly from
the pure Ising values, consistent with that along the critical line of the
XY-Ising model. This suggests that a frustrated XY model or Josephson-junction
array with a zig-zag coupling modulation can provide a physical realization of
the XY-Ising model critical line.
