A note on Todorov surfaces

Let $S$ be a {\em Todorov surface}, {\it i.e.}, a minimal smooth surface of general type with $q=0$ and $p_g=1$ having an involution $i$ such that $S\mathrm{/}i$ is birational to a $K3$ surface and such that the bicanonical map of $S$ is composed with $i\mathrm{.}$ The main result of this paper is that, if $P$ is the minimal smooth model of $S\mathrm{/}i,$ then $P$ is the minimal desingularization of a double cover of $\mathbb P^2$ ramified over two cubics. Furthermore it is also shown that, given a Todorov surface $S$, it is possible to construct Todorov surfaces $S_j$ with $K^2=1,\mathrm{.}\mathrm{.}\mathrm{.},K_S^2-1$ and such that $P$ is also the smooth minimal model of $S_j\mathrm{/}{i}_j,$ where $i_j$ is the involution of $S_j\mathrm{.}$ Some examples are also given, namely an example different from the examples presented by Todorov in \cite{To2}.