  In this paper we consider the four syndrom varieties ${\sf Z}_e^\times$, i.e.
the set of all error locations corresponding to errors of weight $w, 0\leq
w\leq 2$,
  ${\sf Z}_{ns}^\times$ , the set of all {\em non spurious} error locations
corresponding to errors of weight $w, 0\leq w\leq 2$, ${\sf Z}_+^\times $, the
set of all non-spurious error locations corresponding to errors of weight $w,
1\leq w\leq 2$, ${\sf Z}_2^\times $, the set of all non-spurious error
locations corresponding to errors of weight $w= 2$, associated to an up-to-two
errors correcting binary cyclic codes. Denoting $J_\ast:=\mathcal{I}({\sf
Z}_\ast)$, the ideal of these syndrome varieties, ${\sf N}_\ast := {\bf
N}(J_\ast)$ the \GR\ escalier of $J_\ast$ w.r.t. the lex ordering with
$x_1<x_2<z_1<z_2$,
  $\Phi_\ast : {\sf Z}_\ast \to {\sf N}_\ast$ a Cerlienco-Mureddu
correspondence, and $G_*$ a minimal Groebner basis of the ideal $J_\ast$, the
aim of the paper is, assuming to know the structure of the order ideal ${\sf
N}_2$ and a Cerlienco Mureddu Correspondence to deduce with elementary
arguments ${\sf N}_\ast$, $G_\ast$ and $\Phi_\ast$ for $\ast\in\{e,ns,+\}$.
  The tools are Macaulay's trick and Lazard's formulation of Cerlienco-Mureddu
correspondence.
