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Among the physically possible infinitely many material symmetries of all kinds in a two-dimensional space, there exist eight kinds, i.e., the isotropy C_∞ν, hernitropy C_∞, two symmetries C₁ and C₂ in the oblique system, C_1ν and C_2ν in the rectangular system, and C₃ and C_3ν in the trigonal system, that can be characterized in terms of tensors of orders not higher than three. In this paper, the complete and irreducible representations relative to these eight symmetries are established for scalar-, vector-, second-order tensor- and third-order tensor-valued functions of any finite number of vectors, second-order tensors and third-order tensors. These representations allow to obtain, in the case of two-dimensional problems, general invariant forms of the physical laws; in particular, the constitutive equations involving thrid-order tensors.