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A bielliptic surface (or hyperelliptic surface) is a smooth surface with a numerically trivial canonical divisor such that the Albanese morphism is an elliptic fibration. In the first part of this article, we study the structure of bielliptic surfaces over a field of characteristic different from 
$2$ and 
$3$, in order to prove the Shafarevich conjecture for bielliptic surfaces with rational points. Furthermore, we demonstrate that the Shafarevich conjecture does not generally hold for bielliptic surfaces without rational points. In particular, this article completes the study of the Shafarevich conjecture for minimal surfaces of Kodaira dimension 
$0$. In the second part of this article, we study a Néron model of a bielliptic surface. We establish the potential existence of a Néron model for a bielliptic surface when the residual characteristic is not equal to 
$2$ or 
$3$.
