  We compute analytically the density $\varrho_{N,M}(\lambda)$ of Schmidt
eigenvalues, distributed according to a fixed-trace Wishart-Laguerre measure,
and the average R\'enyi entropy $\langle\mathcal{S}_q\rangle$ for reduced
density matrices of entangled random pure states with orthogonal symmetry
$(\beta=1)$. The results are valid for arbitrary dimensions $N=2k,M$ of the
corresponding Hilbert space partitions, and are in excellent agreement with
numerical simulations.
