A method to derive the matrix spectral problems of the Blaszak–Marciniak lattice equations is proposed, and the matrix Lax representations of all the three-field and four-field Blaszak-Marciniak lattice equations are given explicitly. The integrability aspects of a three-field Blaszak–Marciniak lattice equation is studied as an example. To be specific, an integrable lattice hierarchy is constructed based on the matrix spectral problem of this lattice equation, the N-fold Darboux transformation and exact solutions are derived, the solitary wave structures and interaction behaviours of these exact solutions are displayed graphically, and finally the infinitely many conservation laws are listed in a standard way

中文翻译：

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Błaszak-Marciniak 格子方程的矩阵谱问题和可积性方面

提出了一种推导 Blaszak-Marciniak 格子方程矩阵谱问题的方法，并明确给出了所有三场和四场 Blaszak-Marciniak 格子方程的矩阵 Lax 表示。以三场 Blaszak-Marciniak 晶格方程的可积性方面为例进行了研究。具体而言，基于该晶格方程的矩阵谱问题构建可积晶格层次，推导出N次达布变换和精确解，将这些精确解的孤波结构和相互作用行为图形化显示，最后无限多的守恒定律以标准方式列出