Calc. with a self-inverse integer matrix

Dear Axiom-Users,

i am new to axiom. 

I need the solution for the components of a matrix product 
A times B = I.

,,A'' is a self-inverse matrix with integer components, the components of ,,B'' are ploynomial equations containing the components of two vectors which  have their scalar product equal to zero, and one of them is normalized. this leads after all to the system of equations below. 

the problems:

1.) i get:
>> Error detected within library code:
   system does not have a finite number of solutions

however, i was expecting to find at least some more information.

2.) how do i tell ,,axiom'' that the a_ij have to be of type integer?


solve([

(d1+2*n1)*(a11*n1+a12*n2+a13*n3)-a11=1,

(d2+2*n2)*(a21*n1+a22*n2+a23*n3)-a22=1,

(d3+2*n3)*(a31*n1+a32*n2+a33*n3)-a33=1,


(d2+2*n2)*(a11*n1+a12*n2+a13*n3)-a12=0,

(d3+2*n3)*(a11*n1+a12*n2+a13*n3)-a13=0,


(d1+2*n1)*(a21*n1+a22*n2+a23*n3)-a21=0,

(d3+2*n3)*(a21*n1+a22*n2+a23*n3)-a23=0,


(d1+2*n1)*(a31*n1+a32*n2+a33*n3)-a31=0,

(d2+2*n2)*(a31*n1+a32*n2+a33*n3)-a32=0,


n1*n1+n2*n2+n3*n3=1,

d1*n1+d2*n2+d3*n3=0,

a11*a11+a12*a21+a13*a31=1,
a11*a12+a12*a22+a13*a32=0,
a11*a13+a12*a23+a13*a33=0,

a21*a11+a22*a21+a23*a31=0,
a21*a12+a22*a22+a23*a32=1,
a21*a13+a22*a23+a23*a33=0,

a31*a11+a32*a21+a33*a31=0,
a31*a12+a32*a22+a33*a32=0,
a31*a13+a32*a23+a33*a33=1])

what can i do?

by the way, is their a way to define the type ,,self-inverse matrix tith integer components'' in axiom?

thanks,

Rainer

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