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Let two Riemannian metrics g and g on one manifold Mn have the same geodesics (considered as unparameterized curves). Then we can construct invariantly n commuting differential operators of second order. The Laplacian Δg of the metric g is one of these operators. For any x ∈ Mn, consider the linear transformation G of TxMn given by the tensor gIαgαj. If all eigenvalues of G are different at one point of the manifold then they are different at almost every point; the operators are linearly independent and their symbols are functionally independent. If all eigenvalues of G are different at each point of a closed manifold then it can be covered by the n-torus and we can globally separate the variables in the equation Δgf = μf on this torus.