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In this paper, we study the compression of the magnetic particle imaging system matrix for imaging setups in which a field free point is moved along a Lissajous trajectory. We show that a large number of zeros in the simulated transformed system matrix is obtained when orthogonal transforms applied to the spatial domain have only symmetric and antisymmetric basis functions. For measured system matrices, this property only holds approximately, because of noise induced by the scanner hardware. The required symmetry properties are naturally fulfilled by some standard orthogonal transforms such as the type-two discrete cosine transformand the discrete Chebychev transform. However, these transforms are not yet optimal for compressing system matrices, and we propose a new method to obtain better transforms that retain the required symmetry properties.