generalized normal distribution with additional kurtosis parameter

I am looking for a generalized normal distribution with mean $\mu$ and variance $\sigma^2$ but with an additional parameter for the kurtosis to add fat-tails. I can't find it - can anybody help?

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Depends how you want to generalize, but for instance the Student t distribution becomes the standard normal distribution for high degrees of freedom. –  Raskolnikov Dec 21 '10 at 10:33
@Raskolnikov: Ideally I want to let the parameters for mean and variance stay the same but having an additional parameter for kurtosis. –  vonjd Dec 21 '10 at 11:00
That's can be obtained with any distribution with at minimum two free parameters. Just relabel the parameters so that the mean and variance are $\mu$ and $\sigma^2$. Your constraints are too weak to give anything interesting. –  Raskolnikov Dec 21 '10 at 11:22
I cannot follow: The other constraint is that when the third parameter is e.g. 3 (or 0) we are back to the normal distr. –  vonjd Dec 21 '10 at 11:45