  For a system of second order differential equations we determine a nonlinear
connection that is compatible with a given generalized Lagrange metric. Using
this nonlinear connection, we can find the whole family of metric nonlinear
connections that can be associated with a system of SODE and a generalized
Lagrange structure. For the particular case when the system of SODE and the
metric structure are Lagrangian, we prove that the canonic nonlinear connection
of the Lagrange space is the only nonlinear connection which is metric and
compatible with the symplectic structure of the Lagrange space. The metric
tensor of the Lagrange space determines the symmetric part of the nonlinear
connection, while the symplectic structure of the Lagrange space determines the
skew-symmetric part of the nonlinear connection.
