  The differential system for minimal Lagrangian surfaces in a
$2_{\mathbb{C}}$-dimensional, non-flat, complex space form is an elliptic
system defined on the bundle of oriented Lagrangian planes. This is a
6-symmetric space associated with the Lie group SL(3,$\mathbb{C}$), and the
minimal Lagrangian surfaces arise as the primitive maps. Utilizing this
property, we derive the differential algebraic inductive formulas for a pair of
loop algebra $\mathfrak{sl}(3,\mathbb{C})[[\lambda]]$-valued canonical formal
Killing fields. As a result, we give a complete classification of the (infinite
sequence of) Jacobi fields for the minimal Lagrangian system. We also obtain an
infinite sequence of higher-order conservation laws from the components of the
formal Killing fields.
