  A {\em snark} is a cubic cyclically 4-edge connected graph with edge
chromatic number four and girth at least five. We say that a graph $G$ is {\em
odd 2-factored} if for each 2-factor F of G each cycle of F is odd.
  In this paper, we present a method for constructing odd 2--factored snarks.
In particular, we construct two families of odd 2-factored snarks that disprove
a conjecture by some of the authors. Moreover, we approach the problem of
characterizing odd 2-factored snarks furnishing a partial characterization of
cyclically 4-edge connected odd 2-factored snarks. Finally, we pose a new
conjecture regarding odd 2-factored snarks.
