On the symmetric matrix word equation XBX2B3X2BX=A

In this paper we present explicitly the unique positive definite solution of the symmetric word equation XBX2B3X2BX=A over 2 × 2 positive definite matrices. This word equation appeared as a counterexample to the uniqueness of solution conjecture for symmetric word equations: it has multiple positive definite solutions for certain 3 × 3 positive definite matrices A and B.