  We present complete realization of irreducible $A_1 ^{(1)}$-modules at the
critical level in the principal gradation. Our construction uses vertex
algebraic techniques, the theory of twisted modules and representations of Lie
conformal superalgebras. We also provide an alternative Z-algebra approach to
this construction. All irreducible highest weight $A_1 ^{(1)}$-modules at the
critical level are realized on the vector space $M_{\tfrac{1}{2} + \Bbb Z} (1)
^{\otimes 2}$ where $M_{\tfrac{1}{2} + \Bbb Z} (1) $ is the polynomial ring
${\Bbb C}[\alpha(-1/2), \alpha(-3/2), ...]$. Explicit combinatorial bases for
these modules are also given.
