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In this paper, we prove that \"Let $p\\geq0$ and $G$ be a balanced bipartite graph of order $2n$ with minimum degree $\\delta(G)\\geq k$, where $n\\geq 2k-p+2$ and $k\\geq p$. If the number of edges $ e(G)>n(n-k+p-1)+(k+2)(k-p+1), $ then $G$ is $2p$-Hamilton-biconnected except some exceptions.\" Furthermore, this result is used to present two new spectral conditions for a graph to $2p$-Hamilton-biconnected. 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In this paper, we prove that \"Let $p\\geq0$ and $G$ be a balanced bipartite graph of order $2n$ with minimum degree $\\delta(G)\\geq k$, where $n\\geq 2k-p+2$ and $k\\geq p$. If the number of edges $ e(G)>n(n-k+p-1)+(k+2)(k-p+1), $ then $G$ is $2p$-Hamilton-biconnected except some exceptions.\" Furthermore, this result is used to present two new spectral conditions for a graph to $2p$-Hamilton-biconnected. 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