I have a question about indefinite lattices.


QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice,
that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form,
not necessarily unimodular, and $G:=O(\Lambda)$ the group of 
(integer) isometries. Denote the set of all vectors $v\in\Lambda$ such that $v^2=r$ by $S\_r$. I think that it is true (under some 
additional assumptions on rank) that $G$ acts on $S\_r$ with finitely 
many orbits, but I don't know a good reference. 


In this paper we have an argument proving this
for $r=0$: <http://arxiv.org/abs/1208.4626>
(Theorem 3.6), when the rank of a lattice 
is $\geq 7$. 


For unimodular lattices I think there is just
one orbit ("Eichler's theorem"), probably for rank $\geq 5$.


I would be very grateful for any reference
to this result in bigger generality, with
arbitrary $r$ and without unnecessary rank
restrictions.


The question comes from complex geometry: if $M$ is a hyperkahler manifold, there is a canonical non-unimodular integer quadratic form in $H^2(M)$, and its automorphisms are identified (up to finite index) with the mapping class group of $M$. Various geometric questions about $M$ are translated into lattice-theoretic questions about this lattice.



orbits of automorphism group for indefinite lattices
It is in Kneser's book Quadratische Formen. For each r, there are only finitely many classes of representations of r by the lattice. 


