  Bound states in the continuum (BICs) have some unusual properties and
important applications in photonics. A periodic structure sandwiched between
two homogeneous media is the most popular platform for observing BICs and
realizing their applications. Existing studies on BICs assume the periodic
structure has a $C_2$ rotational symmetry about the axis perpendicular to the
periodic layer. It is known that all BICs turn to resonant states with finite
quality factors if the periodic structure is perturbed by a generic
perturbation breaking the $C_2$ symmetry, and a typical BIC continues to exist
if the perturbation keeps the $C_2$ symmetry. We study how typical BICs depend
on generic structural parameters. For a class of BICs with one opening
radiation channel, we show that in the plane of two generic parameters, the
BICs exist continuously as a curve. Consequently, BICs can exist on periodic
structures without the $C_2$ symmetry, and they can be found by tuning a single
structural parameter. The result is established analytically by a perturbation
theory with two independent perturbations and validated by numerical examples.
Our study reveals a much larger family for BICs on periodic structures, and
provides new opportunities for future applications.
