Let $u: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$, $u \in C^2(D)$ where $D$ is the unitary disc, if $u$ satisfies the Laplace equation on $D$, i. e.
$\displaystyle\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$
I want to know if I always can find another solution $v$, $v \in C^{2}(D)$ such that $u$ and $v$ satisfy the Cauchy-Riemann equations, i. e.
$$\displaystyle\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$$
and
$$\displaystyle\frac{\partial v}{\partial x} = -\frac{\partial u}{\partial y}$$