This paper presents the results of specific-heat and magnetization measurements, in particular their field-orientation dependence, on the first discovered heavy-fermion superconductor CeCu$_2$Si$_2$ ($T_{rm c} sim 0.6$ K). We discuss the superconducting gap structure and the origin of the anomalous pair-breaking phenomena, leading e.g., to the suppression of the upper critical field $H_{rm c2}$, found in the high-field region. The data show that the anomalous pair breaking becomes prominent below about 0.15 K in any field direction, but occurs closer to $H_{rm c2}$ for $H parallel c$. The presence of this anomaly is confirmed by the fact that the specific-heat and magnetization data satisfy standard thermodynamic relations. Concerning the gap structure, field-angle dependences of the low-temperature specific heat within the $ab$ and $ac$ planes do not show any evidence for gap nodes. From microscopic calculations in the framework of a two-band full-gap model, the power-law-like temperature dependences of $C$ and $1/T_1$, reminiscent of nodal superconductivity, have been reproduced reasonably. These facts further support multiband full-gap superconductivity in CeCu$_2$Si$_2$.
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