  In general, the system of $2$nd-order partial differential equations made of
the Euler-Lagrange equations of classical field theories are not compatible for
singular Lagrangians. This is the so-called second-order problem. The first aim
of this work is to develop a fully geometric constraint algorithm which allows
us to find a submanifold where the Euler-Lagrange equations have solution, and
split the constraints into two kinds depending on their origin. We do so using
$k$-symplectic geometry, which is the simplest intrinsic description of
classical field theories. As a second aim, the Einstein-Palatini model of
General Relativity is studied using this algorithm.
