2. Expectations of Characteristic Polynomials

Problem 2.1.
Identify whether or not there exists an integrable system (by the usual accepted criteria) underlying \[\mathbb{E}_{A\in U(N)}\left(Z^{2k2h}Z^\prime^{2h}\right),\]
where $Z=\det(I\lambda A)$. 
Problem 2.2.
Consider the expectation given in 2.1. For $h=k$, formulate a conjecture for $k\in\mathbb{R}$ 
Problem 2.5.
How would the conclusions change if one replaces $\mathbb{E}_{A\in U(N)}$ by an average over the other classical groups? 
Problem 2.6.
Calculate $\mathbb{E}\left\frac{Z\prime}{Z}\right^{2k}_{\theta+i\varepsilon}$  is this related to Painlevé?
Cite this as: AimPL: Painleve equations and their applications, available at http://aimpl.org/painleveapp.