  For a scalar evolution equation $u_t=K(t,x,u,u_x,\ldots, u_n), n\geq 2$ the
cohomology spaces $H^{1,s}({\mathcal R}^\infty)$ vanishes for $s\geq 3$ while
the space $H^{1,2}({\mathcal R}^\infty)$ is isomorphic to the space of
variational operators. The cohomology space $H^{1,2}({\mathcal R}^\infty)$ is
also shown to be isomorphic to the space of symplectic operators for $u_t=K$
for which the equation is Hamiltonian. Third order scalar evolution equations
admitting a first order symplectic (or variational) operator are characterized.
The symplectic nature of the potential form of a bi-Hamiltonian evolution
equation is also presented.
