Unit reduction for quadratic norm equations ($\alpha \mapsto \alpha {\epsilon }^n$)

In a real quadratic integer ring, multiplying by a positive norm-one fundamental unit preserves a prescribed norm. Choose an exponent so that a positive representative lies in a half-open interval $[A,A\epsilon )$. Its conjugate is the prescribed norm divided by the representative, so both integer coefficients are bounded on that interval. Enumerating these reduced representatives gives a complete finite list of unit orbits of solutions.