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In the approach given to solve the laplace equation ( With reference to PDE by L Evans ), we first observe that the laplace operator is rotation invariant .i.e., if we rotate the solution ,it still remains a solution. Then, we narrow down by looking for radial functions ( which remain the same after rotation ) and obtain the fundamental solution.

So, my question is:

Is there a (fundamental) solution of the laplace equation which is not radial ?

Srinivas K
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Pick any radial solution and add to it any any harmonic function which is not radial.

Mariano Suárez-Álvarez
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    Bingo ! Another example i found on wiki is $ e^x sin y $. This example comes from the real part of a complex analytic function.Thank you. – Srinivas K Jun 16 '14 at 07:13