If the equation cot⁴ x - 2 cosec² x + a² = 0 has at least one solution, then the sum of all possible integral values of a is equal to

$\begin{array}{l}{\cot }^4\mathrm{x}-2\left(1+{\cot }^2\mathrm{x}\right)+{\mathrm{a}}^2=0\\ \text{ or }{\cot }^4\mathrm{x}-2{\cot }^2\mathrm{x}+{\mathrm{a}}^2-2=0\\ \text{ or }{\left({\cot }^2\mathrm{x}-1\right)}^2=3-{\mathrm{a}}^2\end{array}$

Now, for at least one solution

$\begin{array}{l}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}3-{\mathrm{a}}^2\ge 0\\ \text{ or }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{a}}^2-3\le 0\\ \therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{a}\in [-\sqrt{3},\sqrt{3}]\end{array}$

Integral values are -1, 0, 1.

$\therefore \text{ Sum }=0$