  We call a positive real number $\lambda$ admissible if it belongs to the
Lagrange spectrum and there exists an irrational number $\alpha$ such that
$\mu(\alpha)=\lambda$. Here $\mu(\alpha)$ denotes the Lagrange constant of
$\alpha$ - maximal real number $c$ such that $\forall \varepsilon>0$ the
inequality $|\alpha-\frac{p}{q}|\le\frac{1}{(c-\varepsilon)q^2}$ has infinitely
many solutions for relatively prime $p$ and $q$. In this paper we establish a
necessary and sufficient condition of admissibility of the Lagrange spectrum
element and construct an infinite series of not admissible numbers.
