Let $K$ be a field of characteristic zero and $A_{2}:=A_{2}(K)$ the 2$nd$–Weyl algebra over $K$. We establish a close connection between the maximal left ideals of $A_{2}$ and the simple derivations of $K[X_{1},X_{2}]$.

MAIN THEOREM. Let$d = \partial_1 + \beta\partial_2$be a simple derivation of$K[X_1, X_2]$with$\beta\in K[X_1, X_2]$. Then, there exists$\gamma\in K[X_1, X_2]$such that$d + \gamma$generates a maximal left ideal of$A_2$. More precisely, the following is true:

1. $\deg_{X_2} (\beta) \geq 2$ or $\deg_{X_2} (\beta) = 1$ and $\deg_{X_1} (\partial_2(\beta)) \geq 1$;

2. $d + gX_2$generates a maximal left ideal of$A_2$if$g \in K[X_1]\\{0\}$is such that

1. (a) $g \in -(\partial^2_2(\beta)/2)\mathbb{N}$when$\deg_{X_2} (\beta) \geq 2$,

2. (b) $\deg_{X_1} (g) < \deg_{X_1} (\partial_2(\beta))$when$\deg_{X_2} (\beta) = 1$.

As applications, we obtain large families of concrete examples of cyclic maximal left ideals of $A_2$; such examples have been rather scarce so far.