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Description of two-electron atoms with correct cusp conditions

New sets of functions with arbitrary large finite cardinality are constructed for two-electron atoms. Functions from these sets exactly satisfy the Kato's cusp conditions. The new functions are special linear combinations of Hylleraas- and/or Kinoshita-type terms. Standard variational calculation, leading to matrix eigenvalue problem, can be carried out to calculate the energies of the system. There is no need for optimization with constraints to satisfy the cusp conditions. In the numerical examples the ground state energy of the He atom is considered.