  This paper is devoted to the study of the constraint equations of the
Lovelock gravity theories. In the case of an empty, compact, conformally flat,
time-symmetric, and space-like manifold, we show that the Hamiltonian
constraint equation becomes a generalisation of the $\sigma_k$-Yamabe problem.
That is to say, the prescription of a linear combination of the
$\sigma_k$-curvatures of the manifold. We search solutions in a conformal class
for a compact manifold. Using the existing results on the $\sigma_k$-Yamabe
problem, we describe some cases in which they can be extended to this new
problem. This requires to study the concavity of some polynomial. We do it in
two ways: regarding the concavity of an entire root of this polynomial, which
is connected to algebraic properties of the polynomial; and seeking
analytically a concavifying function. This gives several cases in which a
conformal solution exists. At last we show an implicit function theorem in the
case of a manifold with negative scalar curvature, and find a conformal
solution when the Lovelock theories are close to General Relativity.
