标题： 隐藏对称性，隐藏守恒定律和Nizhnik方程的精确解 显示英文标题

标题： Hidden symmetries, hidden conservation laws and exact solutions of dispersionless Nizhnik equation

摘要： 在（实对称势）无色Nizhnik方程的李子模型中，我们挑选出一个引人注目的子模型，尽管它是唯一的，但它与该方程的最大李不变性代数的一族一般不等价的一维子代数相关，这些子代数由时间变量的一个任意函数参数化。 与上述子模型相关的无色Nizhnik方程的一系列不变解可以用时间变量的一个任意函数和已知的（隐式）一般解关于空间类子模型不变变量的双重四次积分来表示。 这个被挑选出的子模型具有许多其他有趣的性质。 特别是，我们证明了它 是李-显著的，并且它的最大李不变性代数完全定义了它的点对称伪群，这在文献中提供了后一种现象的第二个但更简单的例子。 此外，只有与这个子模型相关的无色Nizhnik方程的隐藏李对称对于找到其精确解是本质的。 利用李约化，我们构造了无粘性Burgers方程和无色Nizhnik方程的新族精确解，以封闭形式或参数形式表示。 我们也详尽地描述了子模型的广义对称性、余对称性和守恒律，这分别给出了无粘性Burgers方程和无色Nizhnik方程相应的非局部和隐藏结构。 显示英文摘要

摘要： Among Lie submodels of the (real symmetric potential) dispersionless Nizhnik equation, we single out a remarkable submodel as such that, despite being the only one, is associated with a family of in general inequivalent one-dimensional subalgebras of the maximal Lie invariance algebra of this equation, which are parameterized by an arbitrary function of the time variable. The wide family of invariant solutions of the dispersionless Nizhnik equation that are related to the above submodel is expressed in terms of an arbitrary function of the time variable and the double quadrature of the well-known (implicit) general solution of the inviscid Burgers equation with respect to a space-like submodel invariant variable. The singled out submodel possesses many other interesting properties. In particular, we show that it is Lie-remarkable, and its maximal Lie invariance algebra completely defines its point symmetry pseudogroup, which provides the second but simpler example of the latter phenomenon in literature. Moreover, only hidden Lie symmetries of the dispersionless Nizhnik equation that are associated with this submodel are essential for finding its exact solutions. Using Lie reductions, we construct new families of exact solutions of the inviscid Burgers equation and the dispersionless Nizhnik equation in closed or parametric form. We also exhaustively described generalized symmetries, cosymmetries and conservation laws of the submodel, which gives the corresponding nonlocal and hidden structures for the inviscid Burgers equation and the dispersionless Nizhnik equation, respectively.