  Let $S^F$ be a $\mathbb{P}$-martingale representing the price of a primitive
asset in an incomplete market framework. We present easily verifiable
conditions on model coefficients which guarantee the completeness of the market
in which in addition to the primitive asset one may also trade a derivative
contract $S^B$. Both $S^F$ and $S^B$ are defined in terms of the solution $X$
to a $2$-dimensional stochastic differential equation: $S^F_t = f(X_t)$ and
$S^B_t:=\mathbb{E}[g(X_1) | \mathcal{F}_t]$. From a purely mathematical point
of view we prove that every local martingale under $\mathbb{P}$ can be
represented as a stochastic integral with respect to the
$\mathbb{P}$-martingale $S := (S^F\ S^B)$. Notably, in contrast to recent
results on the endogenous completeness of equilibria markets, our conditions
allow the Jacobian matrix of $(f,g)$ to be singular everywhere on
$\mathbf{R}^2$. Hence they cover, as a special case, the prominent example of a
stochastic volatility model being completed with a European call (or put)
option.
