I wonder how one can study the integral orthogonal group of a given (symmetric, positive definite) bilinear form like the one described by the following matrix:
$$M=\begin{bmatrix} x_1 & -1 \\ -1 & x_2 & -1 \\ & -1 &\ddots\\ \\ & & & & x_{n-1} &-1 \\ & & & & -1 &x_n \\ \end{bmatrix}$$
I am interested in this specific case where all the $x_i$'s are natural numbers $\geq 2$, with at least one of them being $\neq 2$. The problem of finding a matrix $A$ with integer coefficients such that $$A^TMA=M$$ originates from the study of hypersurface singulraity $f:(\mathbb{C}^3,0)\to (\mathbb{C},0)$ and its Milnor fiber. In particular, one of the restrictions that I can impose further is that the characateristic polynomial of $A$ should be a product of cyclotomic polynomials.
Any suggestion, reference or idea would be very much appreciated!
