  The ageing algebra is a local dynamical symmetry of many ageing systems, far
from equilibrium, and with a dynamical exponent z=2. Here, new representations
for an integer dynamical exponent z=n are constructed, which act non-locally on
the physical scaling operators. The new mathematical mechanism which makes the
infinitesimal generators of the ageing algebra dynamical symmetries, is
explicitly discussed for a n-dependent family of linear equations of motion for
the order-parameter. Finite transformations are derived through the
exponentiation of the infinitesimal generators and it is proposed to interpret
them in terms of the transformation of distributions of spatio-temporal
coordinates. The two-point functions which transform co-variantly under the new
representations are computed, which quite distinct forms for n even and n odd.
Depending on the sign of the dimensionful mass parameter, the two-point scaling
functions either decay monotonously or in an oscillatory way towards zero.
