Abstract
We describe the (alpha, beta)-metrics whose the T-tensor vanishes (T-condition) and the (alpha, beta)-metrics that satisfy the sigma T-condition sigma T-h(ijk)h = 0, where sigma(h) = partial derivative sigma/partial derivative x(h) and sigma is a smooth function on M. These classes have already been obtained by Shen and Asanov in a completely different approach. The Finsler metrics of the first class are Berwaldian, the metrics of the second class are almost regular non-Berwaldian Landsberg metrics.