We develop a model for the gravitational field which is renormalizable
, conformally invariant and integrable in four dimensions. The first t
wo conditions can be easily implemented. However, for the latter condi
tion we must take recourse to fourth-rank geometry where the line elem
ent is defined by a quartic form, ds(4) = G(mu nu lambda rho) dx(mu) d
x(nu) dx(lambda) dx(rho). The simplest Lagrangian which can be constru
cted in this case depends quadratically on a Ricci tensor constructed
only in terms of a connection; therefore a Palatini-like variational p
rinciple is applied. The field equations imply that the fourth-rank me
tric decomposes into a product of a second-rank metric with itself, an
d in this case the geometry becomes Riemannian. The decomposition of t
he fourth-rank metric means our field equations become linear in the R
icci tensor and thus they are amenable for comparison with the Einstei
n field equations. We show that the Einstein field equations are a par
ticular case of our field equations. The field equations are solved in
the spherically symmetric case. The solution contains the Schwarzschi
ld metric and the Kottler metric, corresponding to a massive point par
ticle on a Minkowski and a de Sitter background, respectively.