In this paper we study the PBW filtration on irreducible integrable highest weight representations of affine Kac-Moody algebras
g
^
\widehat {\mathfrak {g}}
. The
n
n
-th space of this filtration is spanned by the vectors
x
1
…
x
s
v
x_1\dots x_s v
, where
x
i
∈
g
^
x_i\in \widehat {\mathfrak {g}}
,
s
≤
n
s\le n
, and
v
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
\mathfrak {g}
of the type
A
1
A_1
we prove our conjecture and derive the fermionic formula for the graded character.