  In this work we classify all the order-two Hypergeometric operators $D$,
symmetric with respect to some $2\times 2$ irreducible matrix-weight $W$ such
that $DP_n=P_n\left(\begin{smallmatrix} \lambda_n&0\\0&\mu_n \end{smallmatrix}
\right)$ with no repetition among the eigenvalues
$\{\lambda_n,\mu_n\}_{n\in\mathbb N_0}$, where $\{P_n\}_{n\in\mathbb N_0}$ is
the (unique) sequence of monic orthogonal polynomials with respect to $W$.
  We obtain, in a very explicit way, a three parameter family of such operators
and weights. We also give the corresponding monic orthongonal polynomials,
their three term recurrence relation and their squared matrix-norms.
