  Starting from an integral projective variety $Y$ equipped with a very ample,
non-special and not-secant defective line bundle $\mathcal{L}$, the paper
establishes, under certain conditions, the regularity of $(Y \times \mathbb
P^2,\mathcal{L}[t])$ for $t\geq 2$. The mildness of those conditions allow to
classify all secant defective cases of any product of $(\mathbb P^1)^{ j}\times
(\mathbb P^2)^{k}$, $j,k \geq 0$, embedded in multidegree at least $(2, \ldots
, 2)$ and $(\mathbb{P}^m\times\mathbb{P}^n\times (\mathbb{P}^2)^k,
\mathcal{O}_{\mathbb{P}^m\times\mathbb{P}^n\times (\mathbb{P}^2)^k} (d,e,t_1,
\ldots, t_k))$ where $d,e \geq 3$, $t_i\geq 2$, for any $n$ and $m$.
