Looking at the typeset result, I'd say you're looking not for cases or aligned, but alignedat (or its standalone version, alignat).
\documentclass{article}
\usepackage{amsmath,amssymb}
\begin{document}
\noindent Standalone:
\begin{alignat*}{99}
\dfrac{\partial u(x,y)}{\partial x}&{}={}& 2&x={}& &\dfrac{\partial v(x,y)}{\partial y} \\
\dfrac{\partial u(x,y)}{\partial y}&{}={}&-2&y={}& -&\dfrac{\partial v(x,y)}{\partial x}
\end{alignat*}
Embedded:
\begin{align*}
\begin{alignedat}{99}
\dfrac{\partial u(x,y)}{\partial x}&{}={}& 2&x={}& &\dfrac{\partial v(x,y)}{\partial y} \\
\dfrac{\partial u(x,y)}{\partial y}&{}={}&-2&y={}& -&\dfrac{\partial v(x,y)}{\partial x}
\end{alignedat}
&&
\forall x \in \mathbb{R}
\end{align*}
\end{document}
The {99} argument is there to satisfy an obtuse requirement of the environment: it specifies the maximum number of columns. I generally don't care to count, so I just make it absurdly large.

It's also possible to have the 2x centered, as tohecz requests, but this entails a small amount of hackery.
\documentclass{article}
\usepackage{amsmath,amssymb}
\begin{document}
\noindent Default
\begin{align*}
\begin{alignedat}{99}
\dfrac{\partial u(x,y)}{\partial x}&{}={}& 2&x={}& &\dfrac{\partial v(x,y)}{\partial y} \\
\dfrac{\partial u(x,y)}{\partial y}&{}={}&-2&y={}& -&\dfrac{\partial v(x,y)}{\partial x}
\end{alignedat}
&&
\forall x \in \mathbb{R}
\end{align*}
$2x$ centered:
\begin{align*}
\begin{alignedat}{99}
\dfrac{\partial u(x,y)}{\partial x}&{}={}&\omit\hfil$2x$\hfil&{}={}& &\dfrac{\partial v(x,y)}{\partial y} \\
\dfrac{\partial u(x,y)}{\partial y}&{}={}& -2y &{}={}& -&\dfrac{\partial v(x,y)}{\partial x}
\end{alignedat}
&&
\forall x \in \mathbb{R}
\end{align*}
\end{document}
Namely, you have to remove the "template" from the column containing 2x and replace it with the necessary glue (not forgetting the $ $, because those are normally included in the template). This entails a slightly different arrangement of columns than before, since now we want to keep the entire monomial together.

alignedinstead ofcases? With a&to start the rows. Welcome to TeX.sx! – egreg Feb 28 '13 at 16:02code. – Ignasi Feb 28 '13 at 16:03