  For a $2$-dimensional non-flat spray we associate a Berwald frame and a
$3$-dimensional distribution that we call the Berwald distribution. The
Frobenius integrability of the Berwald distribution characterises the Finsler
metrizability of the given spray. In the integrable case, the sought after
Finsler function is provided by a closed, homogeneous $1$-form from the
annihilator of the Berwald distribution. We discuss both the degenerate and
non-degenerate cases using the fact that the regularity of the Finsler function
is encoded into a regularity condition of a $2$-form, canonically associated to
the given spray. The integrability of the Berwald distribution and the
regularity of the $2$-form have simple and useful expressions in terms of the
Berwald frame.
