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The paper generalizes results of {\it H. H. Hacisalihoglu} and {\it A. Kh. Amirov} [Dokl. Akad. Nauk, Ross. Akad. Nauk 351, No. 3, 295-296 (1996; Zbl 0895.53038) and Sib. Mat. Zh. 39, No. 4, 1005-1012 (1998; Zbl 0913.53019)] on the existence and uniqueness of a Riemannian metric on a domain in $\bbfR^n$ given prescribed values for some of the components of the Riemann curvature tensor and initial values of the metric and its partial derivatives. The authors establish the construction (existence and uniqueness) of a metric tensor $g_{ij}$ in a semigeodesic coordinate system in a domain $D_n$ in $\bbfR^n$ with certain initial conditions on the metric and its partial derivatives $\frac{ \partial g_{ij}}{\partial x^1}$ on a hypersurface, and prescribed values for the components $R_{1ij1}$ in $D_n$. The result follows from the existence and uniqueness of solutions of systems of first-order ordinary differential equations.
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