This paper is devoted to the study of the constraint equations of the Lovelock gravity theories. In the case of a conformally flat, time-symmetric, and space-like manifold, we show that the Hamiltonian constraint equation becomes a generalisation of the σk-Yamabe problem. That is to say, the prescription of a linear combination of the σk-curvatures of the manifold. We search solutions in a conformal class for a compact manifold. Using the existing results on the σk-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 a 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. Finally 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.