Lie symmetries of a Novikov geometrically integrable equation are found and group-invariant solutions are obtained. Local conservation laws up to second order are established as well as their corresponding conserved quantities. Sufficient conditions for the $L^1$ norm of the solutions to be invariant are presented, as well as conditions for the existence of positive solutions. Two demonstrations for unique continuation of solutions are given: one of them is just based on the invariance of the $L^1$ norm of the solutions, whereas the other is based on well-posedness of Cauchy problems. Finally, pseudo-spherical surfaces determined by the solutions of the equation are studied: all invariant solutions that do not lead to pseudo-spherical surfaces are classified and the existence of an analytic metric for a pseudo-spherical surface is proved using conservation of solutions and well-posedness results.

找到了 Novikov 几何可积方程的李对称，并获得了群不变解。建立了高达二阶的局部守恒定律及其相应的守恒量。充分条件${\mathrm{大号}}^1$给出了解不变的范数，以及正解存在的条件。给出了两个解的唯一延续的演示：其中一个只是基于解的不变性${\mathrm{大号}}^1$解的范数，而另一个基于柯西问题的适定性。最后，研究了由方程解确定的伪球面：对所有不导致伪球面的不变解进行分类，并使用解的守恒证明伪球面的解析度量的存在适定性结果。