Bialynicki-Birula and Charzynski argued that a gravitational wave emitted during the merger of a black hole binary may be approximated by a circularly polarized wave which may in turn trap particles (Bialynicki-Birula and Charzynski 2018 Phys. Rev. Lett. 121 171101). In this paper we consider particle motion in a class of gravitational waves which includes, besides circularly polarized periodic waves (CPP) (Stephani et al 2003 Exact Solutions of Einstein’s Field Equations (Cambridge: Cambridge University Press)), also the one proposed by Lukash V N (1975 Sov. Phys. JETP 40 792; 1974 Zh. Eksp. Teor. Fiz. 67 1594–1608; 1974 Astr. Zh. 51 281; 1976 Nuovo Cimento B 35 208) to study anisotropic cosmological models. Both waves have a seven-parameter conformal symmetry which contains, in addition to the generic five-parameter (broken) Carroll group, also a 6th isometry. The Lukash spacetime can be transformed by a conformal rescaling of time to a perturbed CPP problem. Bounded geodesics, found both analytically and numerically, arise when the Lukash wave is of Bianchi type VI. Their symmetries can also be derived from the Lukash–CPP relation. Particle trapping is discussed.