Darboux integrability of determinant and equations for principal minors

We consider equations that represent a constancy condition for a 2D Wronskian, mixed Wronskian–Casoratian and 2D Casoratian. These determinantal equations are shown to have the number of independent integrals equal to their order—this implies Darboux integrability. On the other hand, the recurrent formulae for the leading principal minors are equivalent to the 2D Toda equation and its semi-discrete and lattice analogues with particular boundary conditions (cut-off constraints). This connection is used to obtain recurrent formulae and closed-form expressions for integrals of the Toda-type equations from the integrals of the determinantal equations. General solutions of the equations corresponding to vanishing determinants are given explicitly while, in the non-vanishing case, they are given in terms of solutions of ordinary linear equations.