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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.