An extension of harmonic functions along fixed direction

Let a function $u(x,y)$ be harmonic in the domain $$D\times V_r=D\times \{y\in {\mathbb{R}}^m:\mathrm{\mid }y\mathrm{\mid }<r\}\subset {\mathbb{R}}^n\times {\mathbb{R}}^m$$ and for each fixed point $x^0$ from some a set $E\subset D$, %which is not embedded in countable association of $N$-sets of $L{h}_0(D)$, the function $u(x^0,y)$, as a function of variable $y$, can be extended to a harmonic function on the whole ${\mathbb{R}}^m$. Then $u(x,y)$ harmonically extends to the domain $D\times $\mathbb{R}$^m$asafunctionofvariables$x$and$y$.