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The object of this paper is to study the properties of flat spacetimes under some conditions regarding the W 2 -curvature tensor. In the first section, several results are obtained on the geometrical symmetries of this curvature tensor. It is shown that in a spacetime with W 2curvature tensor filled with a perfect fluid, the energy momentum tensor satisfying the Einstein's equations with a cosmological constant is a quadratic conformal Killing tensor. It is also proved that a necessary anddoi:10.18514/mmn.2011.332 fatcat:ii437krj7zbyhix4cfkkj7fiui