  We consider the Dirichlet problem for second-order elliptic systems with
constant coefficients. We prove that non-reducible strongly elliptic systems of
this type do not admits non-negatively defined energy functionals of the form
$f\mapsto\int_{D}\varPhi(u_x,v_x,u_y,v_y)\,dxdy$, where $D$ is the domain where
the problem we are interested in is considered, $\varPhi$ is some quadratic
form in $\mathbb R^4$, and $f=u+iv$ is a function in the complex variable. The
proof is based on reducing the system under consideration to a special
(canonical) form, when the differential operator defining this system is
represented as a perturbation of the Laplace operator with respect to two small
real parameters (the canonical parameters of the system under consideration).
