$2\times 2$ Laguerre-type differential operator with triangular eigenvalue

In this paper, we present a comprehensive account of all Laguerre-type differential operators $D$ that are symmetric with respect to a $2\times 2$ irreducible weight $W$ on the interval $(0, \infty)$. These operators are associated with monic orthogonal polynomials ${P_n}$, which satisfy the equation $DP_n = P_n\Delta_n$ for a certain lower triangular eigenvalue $\Delta_n$. We introduce three distinct families of operators and weights, each characterized by explicit expressions depending on two or three parameters, along with a new expression based on a single parameter.