Nonuniqueness of a symmetric word equation in two-by-two positive definite matrices

The symmetric word equation $XBX^{12}BX = P$, with
$B = \begin{pmatrix} 1 & 4 \\ 4 & 17 \end{pmatrix}$ and an explicit integer
matrix $P$, has at least three distinct real symmetric positive definite
$2 \times 2$ solutions. At the solution $X_0 = \operatorname{diag}(3, 1)$ the
Jacobian determinant of the word map in symmetric-matrix coordinates is exactly
$-11785057051824$, so Armstrong and Hillar's degree theorem already gives a
second solution, disproving their conjecture that symmetric word equations in
$2 \times 2$ positive definite matrices have unique solutions. Separate integer
interval-arithmetic certificates establish two further roots without using the
degree theorem. A factored Jacobian also gives a sharp classification within
the family $XBX^rBX$ over real symmetric positive definite $2 \times 2$ matrices: the
equation has exactly one solution for all $B$ and $P$ when $1 \le r \le 6$,
whereas for every integer $r \ge 7$ some $B$ and $P$ give more than one.