Fundamental solutions of a Pell equation

Problem 4.2 (Dujella, 2026). Is there any positive integer $k$, $k \neq 33539$, such that the Pellian equation \[ x^2-(k^2-1)y^2=2-k^2 \] has more than four fundamental solutions?

Remark. For $k=33539$ there are six fundamental solutions: $(\pm 1,1)$, $(\pm 669941,20)$, $(\pm 4326401,129)$.