Generalized Fruit Diophantine equation over number fields

Let $K$ be a number field and ${\mathcal{O}}_K$ be the ring of integers of $K$. In this article, we study the solutions of the generalized fruit Diophantine equation $a{x}^d-y^2-z^2+xyz-c=0$ over $K$, where $d\ge 3$ is an integer and $a,c\in {\mathcal{O}}_K\setminus \{0\}$. Subsequently, we provide explicit values of square-free integers $t$ such that the equation $a{x}^d-y^2-z^2+xyz-c=0$ has no solution $(x_0,y_0,z_0)\in {\mathcal{O}}_{\mathbb{Q}(\sqrt{t})}^3$ with $2\mathrm{\mid }{x}_0$, and demonstrate that the set of all such square-free integers $t$ with $t\ge 2$ has density exactly $\frac{1}{6}$. As an application, we construct infinitely many elliptic curves $E$ defined over number fields $K$ having no integral point $(x_0,y_0)\in {\mathcal{O}}_K^2$ with $2\mathrm{\mid }{x}_0$.