  We study pure Lovelock vacuum and perfect fluid equations for Kasner-type
metrics. These equations correspond to a single $N$th order Lovelock term in
the action in $d=2N+1,\,2N+2$ dimensions, and they capture the relevant
gravitational dynamics when aproaching the big-bang singularity within the
Lovelock family of theories. Pure Lovelock gravity also bears out the general
feature that vacuum in the critical odd dimension, $d=2N+1$, is kinematic; i.e.
we may define an analogue Lovelock-Riemann tensor that vanishes in vacuum for
$d=2N+1$, yet the Riemann curvature is non-zero. We completely classify
isotropic and vacuum Kasner metrics for this class of theories in several
isotropy types. The different families can be characterized by means of certain
higher order 4th rank tensors. We also analyze in detail the space of vacuum
solutions for five and six dimensional pure Gauss-Bonnet theory. It possesses
an interesting and illuminating geometric structure and symmetries that carry
over to the general case. We also comment on a closely related family of
exponential solutions and on the possibility of solutions with complex Kasner
exponents. We show that the latter imply the existence of closed timelike
curves in the geometry.
