The majority of answers already given have already provided some insight into the obvious methodological flaws, so I will not dwell on that here. Instead, I'll provide a few practical options on how to treat this data given that it's already collected and despite the flawed question. There are a few ways to handle this. You could consider marking responses that did not meet your definition of a "valid responses" by treating the entire question as missing and then following any number of practices for handling item-nonresponse such as those discussed here.

You might also consider scaling each response so that the percentages add to 100. Assuming each response is recorded as a percentage, this can be done by re-coding each original response $y_{{old}_{ij}}$ $(j=1,2,3,4)$ of the 4 sub-components to your question (i.e. artistic activity, government support, private pension, activities not related with arts) into a new response $y_{{new}_{ij}}$ as follows:

\begin{eqnarray*} y_{new_{ij}} & = & \begin{cases} 0 & ,\,\text{for}\,\sum_{j=1}^{4}y_{old_{ij}}=0\\ \frac{y_{old_{ij}}}{\sum_{j=1}^{4}y_{old_{ij}}} & ,\,\text{for}\,\sum_{j=1}^{4}y_{old_{ij}}>0\\ Missing & ,\,Otherwise \end{cases} \end{eqnarray*} So for example, say you had a respondent $i$ who answered as follows:

 A. (i=1) Artistic activity:  10% 
 B. (i=2) Government support: 0% 
 C. (i=3) Private pension: 30% 
 D. (i=1) Activities not related with arts:  40%

Then you'd recode as follows:

\begin{eqnarray*} y_{new_{i1}} & = & \frac{10}{10+0+30+40}=\frac{10}{80}=12.5\%\\ y_{new_{i2}} & = & \frac{0}{10+0+30+40}=\frac{0}{80}=00.0\%\\ y_{new_{i3}} & = & \frac{30}{10+0+30+40}=\frac{30}{80}=37.5\%\\ y_{new_{i4}} & = & \frac{40}{10+0+30+40}=\frac{40}{80}=50.0\% \end{eqnarray*}

Note that all the new percentages now add to 100%. Whatever you do, please be sure you make any transformations very clear when reporting your results.

The majority of answers already given have already provided some insight into the obvious survey methodological flaws, so I will not dwell on that here. Instead, I'll provide a few practical options on how to treat this data given that it's already collected and despite the flawed question. There are a few ways to handle this. You could consider marking responses that did not meet your definition of a "valid responses" by treating the entire question as missing and then following any number of practices for handling item-nonresponse such as those discussed here.

You might also consider scaling each response so that the percentages add to 100. Assuming each response is recorded as a percentage, this can be done by re-coding each original response $y_{{old}_{ij}}$ $(j=1,2,3,4)$ of the 4 sub-components to your question (i.e. artistic activity, government support, private pension, activities not related with arts) into a new response $y_{{new}_{ij}}$ as follows:

\begin{eqnarray*} y_{new_{ij}} & = & \begin{cases} 0 & ,\,\text{for}\,\sum_{j=1}^{4}y_{old_{ij}}=0\\ \frac{y_{old_{ij}}}{\sum_{j=1}^{4}y_{old_{ij}}} & ,\,\text{for}\,\sum_{j=1}^{4}y_{old_{ij}}>0\\ Missing & ,\,Otherwise \end{cases} \end{eqnarray*} So for example, say you had a respondent $i$ who answered as follows:

 A. (i=1) Artistic activity:  10% 
 B. (i=2) Government support: 0% 
 C. (i=3) Private pension: 30% 
 D. (i=1) Activities not related with arts:  40%

Then you'd recode as follows:

\begin{eqnarray*} y_{new_{i1}} & = & \frac{10}{10+0+30+40}=\frac{10}{80}=12.5\%\\ y_{new_{i2}} & = & \frac{0}{10+0+30+40}=\frac{0}{80}=00.0\%\\ y_{new_{i3}} & = & \frac{30}{10+0+30+40}=\frac{30}{80}=37.5\%\\ y_{new_{i4}} & = & \frac{40}{10+0+30+40}=\frac{40}{80}=50.0\% \end{eqnarray*}

Note that all the new percentages now add to 100%. Whatever you do, please be sure you make any transformations very clear when reporting your results and I think @rolando2 provided some excellent advice on how to perform some sensitivity analyses to see how transformations like these might affect your conclusions.