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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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    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.2010-12-21

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I think Pearson type VII distribution might be what your are looking for. You will need to reparametrize it, but wikipedia page has the necessary formulas.

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    @vonjd, you're welcome. Thank helpful people who mentioned this distribution in kurtosis wikipedia page, that is how I found it. If by log-pearson you mean distribution of random variable whose log is pearson VII, you can derive its distribution function similarly to log-normal. Hence no readily available references.2010-12-21