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It is known that
$$ A_1(x_1, x_2) = \partial \varphi(x_1, x_2)/\partial x_1, $$ $$ A_2(x_1, x_2) = \partial \varphi(x_1, x_2)/\partial x_2 $$ holds if and only if $$ \partial A_1/\partial x_2 = \partial A_2/\partial x_1. $$

What would be criteria on $A_i$ in a more general case: $$ A_i(x) = F_i(B_j(x), \partial B_k(x)/\partial x) $$ where $x = (x_1, ..., x_n)$ and number of functions $A_i$ is bigger than the number of functions $B_k$.

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