Nuclear magnetic resonance measurements were performed on CeCu$_{2}$Si$_{2}$ in the presence of a magnetic field close to the upper critical field $mu_{0} H_{rm c2}$ in order to investigate its superconducting (SC) properties near pair-breaking fields. In lower fields, the Knight shift and nuclear spin-lattice relaxation rate divided by temperature $1/T_1T$ abruptly decreased below the SC transition temperature $T_{rm c}(H)$, a phenomenon understood within the framework of conventional spin-singlet superconductivity. In contrast, $1/T_1T$ was enhanced just below $T_{rm c}(H)$ and exhibited a broad maximum when magnetic fields close to $mu_0H_{rm c2}(0)$ were applied parallel or perpendicular to the $c$ axis, although the Knight shift decreased just below $T_{rm c}(H)$. This enhancement of $1/T_1T$, which was recently observed in the organic superconductor $kappa$-(BEDT-TTF)$_{2}$Cu(NCS)$_{2}$, suggests the presence of high-density Andreev bound states in the inhomogeneous SC region, a hallmark of the Fulde-Ferrell-Larkin-Ovchinnikov phase.
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