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We study relations between functions on the cotangent bundle of a spacetime which are constants of motion for geodesics and functions on the odd-dimensional phase space conserved by the Reeb vector fields of geometrical structures generated by the metric and an electromagnetic field. 2010 Mathematics Subject Classification: primary 70H40; secondary 70H45, 70H33, 70G45, 58A20.doi:10.5817/am2015-5-297 fatcat:55353nqjevep3p2sanoupppuqa