  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
formulas 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
formulas 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.
