  In this note, we will do analysis of accessible singular points for a
polynomial Hamiltonian system obtained by taking a double covering of the
Painlev\'e I equation. We will show that this system passes the Painlev\'e
$\alpha$-test for all accessible singular points $P_i \ (i=1,2,3)$. We note its
holomorphy condition of the first Painlev\'e system.
