  A periodic structure sandwiched between two homogeneous media can support
bound states in the continuum (BICs) that are valuable for many applications.
It is known that generic BICs in periodic structures with an up-down mirror
symmetry and an in-plane inversion symmetry are robust with respect to
structural perturbations that preserve these two symmetries. For
two-dimensional (2D) structures with one periodic direction and the up-down
mirror symmetry (without the in-plane inversion symmetry), it was recently
established that some scalar BICs can be found by tuning a single structural
parameter. In this paper, we analyze vectorial BICs in such 2D structures, and
show that a typical vectorial BIC with nonzero wavenumbers in both the
invariant and the periodic directions can only be found by tuning two
structural parameters. Using an all-order perturbation method, we prove that
such a vectorial BIC exists as a curve in the 3D space of three generic
parameters. Our theory is validated by numerical examples involving periodic
arrays of dielectric cylinders. The numerical results also illustrate the
conservation of topological charge when structural parameters are varied, if
both BICs and circularly polarized states (CPSs) are included. Our study
reveals a fundamental property of BICs in periodic structure and provides a
systematically approach for finding BICs in structures with less symmetry.
