A differential equation without a solution

A direct construction is given of a function $f(x_1, x_2)\in C^\infty$, such that the equation
$$ \frac{\partial u}{\partial x_1}+ix_1^{2k-1}\frac{\partial u}{\partial x_2}=f $$
has no solution in any neighborhood of the origin; the function $f$ and all its derivatives vanish for $x_1=0$.