Representative systems of exponentials and the Cauchy problem for partial differential equations with constant coefficients

We consider the Cauchy problem with respect to $z_2$ for a homogeneous linear partial differential equation with constant coefficients in two independent variables $z_1,z_2 \in \mathbb C$. We show that the relative smoothness with respect to $z_1$ and $z_2$ of analytic and ultradifferentiable solutions of the Cauchy problem depends essentially on the value of $\rho_2$ and, as a rule, is completely determined by it. We also obtain rather general uniqueness theorems and find conditions which guarantee that the particular solution constructed depends both continuously and linearly on the initial functions.