On a class of almost regular Landsberg metrics

There is a long existing "unicorn" problem in Finsler geometry: whether or not any Landsberg metric is a Berwald metric? Some classes of metrics were studied in the past and no regular non-Berwaldian Landsberg metric was found. However, if the metric is almost regular (allowed to be singular in some directions), some non-Berwaldian Landsberg metrics were found in the past years. All of them are composed by Riemannian metrics and 1-forms. This motivates us to find more almost regular non-Berwaldian Landsberg metrics in the class of general (alpha, beta)-metrics. In this paper, we first classify almost regular Landsberg general (alpha, beta)-metrics into three cases and prove that those regular metrics must be Berwald metrics. By solving some nonlinear PDEs, some new almost regular Landsberg metrics are constructed which have not been described before.