Matrix bispectrality of full rank one algebras

We consider a noncommutative version of the bispectral problem proposed originally by Grünbaum. The latter had number unexpected connections with soliton theory and special functions interest in Mathematical Physics. take an algebraic route to describe symmetry structure algebras operators using notion full rank one algebra. By this concept, we provide general framework derive method verify if such matrix polynomial sub-algebra is bispectral. give two examples illustrating method. In first one, eigenvalue be scalar-valued, whereas second assume it matrix-valued. former example put forth Pierce decomposition that As byproduct, answer positively conjecture Grünbaum concerning certain associated problem.