  We study the distribution of the supremum of the Airy process with $m$
wanderers minus a parabola, or equivalently the limit of the rescaled maximal
height of a system of $N$ non-intersecting Brownian bridges as $N\to\infty$,
where the first $N-m$ paths start and end at the origin and the remaining $m$
go between arbitrary positions. The distribution provides a $2m$-parameter
deformation of the Tracy--Widom GOE distribution, which is recovered in the
limit corresponding to all Brownian paths starting and ending at the origin.
  We provide several descriptions of this distribution function: (i) A Fredholm
determinant formula; (ii) A formula in terms of Painlev\'e II functions; (iii)
A representation as a marginal of the KPZ fixed point with initial data given
as the top path in a stationary system of reflected Brownian motions with
drift; (iv) A characterization as the solution of a version of the
Bloemendal--Virag PDE (arXiv:1011.1877, arXiv:1109.3704) for spiked
Tracy--Widom distributions; (v) A representation as a solution of the KdV
equation. We also discuss connections with a model of last passage percolation
with boundary sources.
