New examples of indefinite hyper-Kähler symmetric spaces

Following the approach to pseudo-Riemannian symmetric spaces developed in [I. Kath, M. Olbrich, On the structure of pseudo-Riemannian symmetric spaces, arXiv:math.DG/0408249, 2004] we exhibit examples of indefinite hyper-Kähler symmetric spaces with non-abelian holonomy. Moreover, we classify indecomposable hyper-Kähler symmetric spaces whose metric has signature (4,4n)(4,4n). Such spaces exist if and only if n∈{0,1,3}n∈{0,1,3}.