Split date periods by months and add rows in case of necessity

I have a table:

Name| Start     | Finish    |
----|-----------|-----------|
A   |2015-01-22 |2015-02-04 |
B   |2015-01-02 |2015-01-10 |
A   |2015-01-22 |2015-02-14 |
B   |2015-01-02 |2015-02-10 |

I need to break periods by months. If a period starts in one month and ends in the next one then I need to split it into two periods. If a period starts and ends at the same month then it should be as it is. Let's assume period cannot contain more than one 1st day of the month. In other words, each line can be splitted for not more than two lines. Finish (end of the period) is always bigger than Start.

That's what I want to get:

Name| Start     | Finish    |
----|-----------|-----------|
A   |2015-01-22 |2015-01-31 |
A   |2015-02-01 |2015-02-04 |
A   |2015-01-22 |2015-01-31 |
A   |2015-02-01 |2015-02-14 |
B   |2015-01-02 |2015-01-10 |
B   |2015-01-02 |2015-01-31 |
B   |2015-02-01 |2015-02-10 |

The order of output rows isn't a matter.

Here is a code for the table:

Name = c("A", "B", "A", "B")
Start = c(as.Date("2015-01-22"), as.Date("2015-01-02"), as.Date("2015-01-22"), as.Date("2015-01-02"))
Finish = c(as.Date("2015-02-04"), as.Date("2015-01-10"), as.Date("2015-02-14"), as.Date("2015-02-10"))
df = data.frame(Name, Start, Finish)

Any suggestion how it can be done?