  In this paper we study the PBW filtration on irreducible integrable highest
weight representations of affine Kac-Moody algebra $\gh$. The $n$-th space of
this filtration is spanned with the vectors $x_1... x_s v$, where $x_i\in\gh$,
$s\le n$ and $v$ is a highest weight vector. For the vacuum module we give a
conjectural description of the corresponding adjoint graded space in terms of
generators and relations. For $\g$ of the type $A_1$ we prove our conjecture
and derive the fermionic formula for the graded character.
