  In this paper we consider $ X(\bar\varphi)$ anisotropic symmetric space $
2\pi$ of periodic functions of $m$ variables, in particular, the generalized
Lorentz space $L_{\bar{\psi},\bar{\tau}}^{*}(\mathbb{T}^{m})$ and
Nikol'skii--Besov's class $S_{X(\bar{\varphi}),\bar{\theta}}^{\bar r}B$. The
article proves an embedding theorem for the Nikol'skii - Besov class in the
generalized Lorentz space and establishes an upper bound for the best
approximations by trigonometric polynomials with harmonic numbers from the
hyperbolic cross of functions from the class
$S_{X(\bar{\varphi}),\bar{\theta}}^{\bar r}B$.
