  In this paper we revisit a discrete spectral problem which was proposed by
Ragnisco and Tu in 1989, as a second discretization of the ZS-AKNS spectral
problem. We show that the spectral problem corresponds to a bidirectional
discretization of the derivative of two wave functions $\phi_{1,x}$ and
$\phi_{2,x}$. As a connection with higher dimensional systems, the spectral
problem and a related hierarchy can be derived from Lax triads of the
differential-difference KP hierarchy via a symmetry constraint. Isospectral and
nonisospectral flows derived from the spectral problem compose a Lie algebra.
By considering its infinite dimensional subalgebras and continuum limit of
recursion operator, three semi-discrete AKNS hierarchies are constructed.
