In accordance to the classical theory of differential equations we define the Vecua--Bernoulli and Vecua--Riccati equations in the space of functions of two complex variables $z$ and $\bar z$ (where the conjugate of unknown function w is a novelty) and give possibilities for solving of these equations using basic I.\,N. Vecua's equations with analytic coefficients $A(z,\bar z)$, $B(z,\bar z)$ which are defined by their convergent areolar series in a domain $D_{x\circ y}$.